parabolic Allen–Cahn equation in codimension one [37]), a key ingredient in
establishing the desired monotonicity result will be bounding the discrepancy
function

(6.3) $\xi_{t}:=\epsilon|d\alpha_{t}|-\frac{1-|u_{t}|^{2}}{2\epsilon}$

along the flow.

As in the stationary case [50, Section 3], it is straightforward to check that
solutions of (6.1) satisfy the following identities: letting

$\omega_{t}:=d\alpha_{t}$

and

$\psi(u_{t},\nabla_{t})(e_{j},e_{k}):=2\langle
i\nabla_{e_{j}}u,\nabla_{e_{k}}u\rangle,$

we have

(6.4)
$\epsilon^{2}(\partial_{t}+\Delta_{H})\omega_{t}=\psi(u_{t},\nabla_{t})-|u_{t}|^{2}\omega_{t},$

from which one obtains the parabolic Bochner identity

(6.5)
$-\epsilon^{2}(\partial_{t}+d^{*}d)\frac{1}{2}|\omega_{t}|^{2}=|u_{t}|^{2}|\omega_{t}|^{2}+\epsilon^{2}|D\omega_{t}|^{2}-\langle\psi(u_{t},\nabla_{t}),\omega_{t}\rangle+\epsilon^{2}\mathcal{R}_{2}(\omega_{t},\omega_{t}),$

where $\mathcal{R}_{2}$ denotes the Weitzenböck curvature operator for two-
forms. Also,

(6.6)
$-(\partial_{t}+d^{*}d)\frac{1}{2}|u_{t}|^{2}=|\nabla_{t}u_{t}|^{2}-\frac{1}{2\epsilon^{2}}(1-|u_{t}|^{2})|u_{t}|^{2}.$

######  Remark 6.1.

It is an easy consequence of (6.6) and the parabolic maximum principle that
$|u_{t}|\leq 1$ for all $t>0$, for initial sections $u_{0}$ satisfying
$|u_{0}|\leq 1$.

By a combination of (6.5) and (6.6), similarly to [50], we find that the
discrepancy function in (6.3) satisfies the weak differential inequality

(6.7) $-(\partial_{t}+d^{*}d+\epsilon^{-2}|u_{t}|^{2})\xi_{t}\geq-
C_{0}(M)\epsilon|\omega_{t}|.$

Equivalently, writing

$\bar{\xi}_{t}:=e^{-C_{0}t}\xi_{t},$

we have

(6.8) $-(\partial_{t}+d^{*}d+\epsilon^{-2}|u_{t}|^{2})\bar{\xi}_{t}\geq-
C_{0}e^{-C_{0}t}\frac{1-|u_{t}|^{2}}{2\epsilon}\geq-\frac{C_{0}}{2\epsilon}(1-|u_{t}|^{2}).$

Now, let $K(t,x,y)$ be the heat kernel of $M$, so that

$(\partial_{t}+d^{*}d)K(t,\cdot,y)=0\quad\text{and}\quad\lim_{t\to
0}K(t,\cdot,y)=\delta_{y}.$

Define then

$\varphi(t,x):=\int_{M}K(t,x,y)|\xi_{0}|(y)\,dy$

and

$\psi(t,x):=\int_{0}^{t}\int_{M}K(t-s,x,y)\frac{C_{0}}{2\epsilon}(1-|u_{s}|^{2})(y)\,dy\,ds.$

Thus, $\varphi$ is the nonnegative solution of the heat equation
$-(\partial_{t}+d^{*}d)\varphi=0$, with initial condition
$\varphi(0,x)=|\xi_{0}(x)|$. By Duhamel’s principle, $\psi$ is the nonnegative
solution of the inhomogeneous heat equation

$-(\partial_{t}+d^{*}d)\psi=-\frac{C_{0}}{2\epsilon}(1-|u_{t}|^{2}),$

with boundary data $\psi(0,x)=0.$ In particular, it follows from (6.8) that

(6.9)
$-(\partial_{t}+d^{*}d+\epsilon^{-2}|u_{t}|^{2})(\bar{\xi}_{t}-\psi_{t}-\varphi_{t})\geq\frac{|u_{t}|^{2}}{\epsilon^{2}}(\varphi_{t}+\psi_{t})\geq
0,$

while $\xi_{0}-\psi_{0}-\varphi_{0}=\xi_{0}-|\xi_{0}|\leq 0$. Hence, the
parabolic maximum principle (for continuous weak solutions) implies the
pointwise bound

(6.10) $\bar{\xi}_{t}\leq\varphi_{t}+\psi_{t}.$

We now use the following well-known asymptotics for the heat kernel on a
compact manifold (see, e.g., [15, Chapter VI]).

######  Lemma 6.2.

We have $(4\pi t)^{n/2}e^{d(x,y)^{2}/4t}K(t,x,y)\to 1$ as $t\to 0^{+}$,
uniformly in $x,y\in M$.

In particular, since $|K(t,x,y)|\leq C(\tau,M)$ for any $t\geq\tau>0$, one has

$\int_{M}K(t,x,y)^{p}\,dy\leq C(p)\max\\{t^{(1-p)n/2},1\\}.$

Since $|u_{t}|\leq 1$ by Remark 6.1, we have automatically

$\Big{\|}\frac{1}{\epsilon}(1-|u_{t}|^{2})\Big{\|}_{L^{\infty}(M)}\leq\frac{1}{\epsilon}\quad\text{and}\quad\Big{\|}\frac{1}{\epsilon}(1-|u_{t}|^{2})\Big{\|}_{L^{2}(M)}\leq
2\sqrt{\Lambda}$

for every $t$, and interpolating we see that

$\Big{\|}\frac{1}{\epsilon}(1-|u_{t}|^{2})\Big{\|}_{L^{q}(M)}\leq
C(M,\Lambda)\epsilon^{(2-q)/q}$

for $2\leq q\leq\infty$. It follows that, for $p\in(1,\frac{n}{n-2})$ with
Hölder conjugate $q$,

$\displaystyle\psi(t,x)$
$\displaystyle\leq\int_{0}^{t}\|K(t-s,x,y)\|_{L^{p}(M)}C\epsilon^{(2-q)/q}\,ds$
$\displaystyle\leq
C\epsilon^{(2-q)/q}\int_{0}^{t}(t-s)^{\frac{n(1-p)}{2p}}\,ds$
$\displaystyle\leq
C(p,M,\Lambda)\epsilon^{(2-q)/q}\Big{(}\frac{n}{2p}-\frac{n-2}{2}\Big{)}^{-1}t^{\frac{n}{2p}-\frac{n-2}{2}},$

provided that $q\geq 2$. In particular, taking $p:=\frac{n-1}{n-2}\text{ and
}q:=n-1$, we arrive at an estimate of the form

$\psi(t,x)\leq
C_{1}(M,\Lambda)\epsilon^{\frac{3-n}{n-1}}t^{\frac{n-2}{2(n-1)}}.$

Now, let

$\eta_{t}:=\bar{\xi}_{t}-\varphi_{t}\leq\psi_{t}\leq
C_{1}\epsilon^{\frac{3-n}{n-1}}t^{\frac{n-2}{2(n-1)}},$

and setting

$f_{t}:=\eta_{t}-C_{1}\epsilon^{\frac{3-n}{n-1}}t^{\frac{n-2}{2(n-1)}}(1-|u_{t}|^{2}),$

note that

$f_{t}\leq C_{1}\epsilon^{\frac{3-n}{n-1}}t^{\frac{n-2}{2(n-1)}}|u_{t}|^{2}$

pointwise. On the other hand, recalling (6.8), note that $f_{t}$ satisfies

$\displaystyle-(\partial_{t}+d^{*}d)f_{t}$
$\displaystyle\geq\frac{|u_{t}|^{2}}{\epsilon^{2}}\bar{\xi}_{t}-\frac{C_{0}}{2\epsilon}(1-|u_{t}|^{2})+C_{1}\epsilon^{\frac{3-n}{n-1}}t^{\frac{n-2}{2(n-1)}}(2|\nabla_{t}u_{t}|^{2}-\epsilon^{-2}(1-|u_{t}|^{2})|u_{t}|^{2})$
$\displaystyle\geq\frac{|u_{t}|^{2}}{\epsilon^{2}}f_{t}-\frac{C_{0}}{2\epsilon}(1-|u_{t}|^{2}),$

and since $|u_{t}|^{2}\geq
c\epsilon^{\frac{n-3}{n-1}}t^{\frac{2-n}{2(n-1)}}f_{t}$, it follows that on
$\\{f>0\\}$ we have

$-(\partial_{t}+d^{*}d)f_{t}\geq\epsilon^{-2}(c\epsilon^{\frac{n-3}{n-1}}t^{\frac{2-n}{2(n-1)}}f_{t}^{2}-C_{0}\epsilon).$

Note that $f_{0}=\xi_{0}-|\xi_{0}|\leq 0$. For any $\tau>0$, if $f$ has a
positive maximum on $[0,\tau]\times M$ at some point $(t,x)$ with $t>0$, then
the last weak subequation implies that here

$\displaystyle
c\epsilon^{\frac{n-3}{n-1}}t^{\frac{2-n}{2(n-1)}}f_{t}^{2}-C_{0}\epsilon\leq
0,$

or equivalently

$\displaystyle f_{t}\leq C\epsilon^{\frac{1}{n-1}}t^{\frac{n-2}{4(n-1)}}\leq
C\epsilon^{\frac{1}{n-1}}\tau^{\frac{n-2}{4(n-1)}}.$

The same inequality holds then on all of $[0,\tau]\times M$. Since $\tau$ was
arbitrary, we obtain

$\displaystyle f_{t}\leq C\epsilon^{\frac{1}{n-1}}t^{\frac{n-2}{4(n-1)}}$

for all $t\geq 0$. Recalling the definitions of $f$, $\eta$, $\bar{\xi}$ and
$\varphi$, the preceding estimate tells us that

(6.11) $\xi_{t}\leq
Ce^{Ct}\Big{(}\varphi_{t}+\epsilon^{\frac{1}{n-1}}+\epsilon^{\frac{2}{n-1}}\frac{1-|u|^{2}}{\epsilon}\Big{)},$

where $\varphi$ is the solution of the heat equation with initial data
$\varphi_{0}=|\xi_{0}|$, for a constant $C=C(M,\Lambda)$. Finally, noting that

$\varphi_{t}\leq C\|\xi_{0}\|_{L^{1}(M)}\leq C(M,\Lambda)\quad\text{for $t\geq
1,$}$

it follows from the above that

(6.12) $\xi_{t}\leq
Ce^{Ct}(1+\epsilon^{\frac{2}{n-1}}\sqrt{e_{\epsilon}(u_{t},\nabla_{t})})\quad\text{for
$t\geq 1$}.$

###  6.2. Huisken-type monotonicity and $\bm{(n-2)}$-energy-density bounds
along the flow

As above, let $(u_{t},\nabla_{t})$ be a solution of the gradient flow with
$E_{\epsilon}(u_{0},\nabla_{0})\leq\Lambda$ and $|u_{0}|\leq 1$. Mimicking the
computations leading to Huisken’s monotonicity for the mean curvature flow
[35], let us introduce $h(t,x)$, a positive solution of the _backward_ heat
equation

$\partial_{t}h=d^{*}dh$

on $[0,T)\times M$, with $\int_{M}h=1$. Write
$e_{t}:=e_{\epsilon}(u_{t},\nabla_{t})$ to lighten the notation and set

$\Phi_{h}(t):=\int_{M}he_{t}.$

Integration by parts combined with the gradient flow equations allows us to
deduce that

$\displaystyle\Phi_{h}^{\prime}(t)$
$\displaystyle=\int_{M}(\partial_{t}he_{t}+h\partial_{t}e_{t})$
$\displaystyle=\int_{M}[(d^{*}dh)e_{t}+h(2\langle\nabla\dot{u}-i\dot{\alpha}u,\nabla
u\rangle+2\epsilon^{2}\langle
d\dot{\alpha},d\alpha\rangle-\epsilon^{-2}(1-|u|^{2})\langle
u,\dot{u}\rangle)]$ $\displaystyle=\int_{M}[\langle
dh,de_{t}\rangle-2h(|\dot{u}|^{2}+\epsilon^{2}|\dot{\alpha}|^{2})-2(\langle\nabla_{dh}u,\dot{u}\rangle+\epsilon^{2}d\alpha(dh,\dot{\alpha}))]$

(where we dropped the subscript $t$ from $u_{t}$, $\alpha_{t}$, $\dot{u}_{t}$,
and $\dot{\alpha}_{t}$). Next, recall from [50, Section 4] the stress-energy
tensor

$T_{\epsilon}(u,\nabla):=e_{\epsilon}(u,\nabla)g-2\nabla u^{*}\nabla
u-2\epsilon^{2}d\alpha^{*}d\alpha,$

and note (cf. [50, Section 4]) that we have the identities

$\displaystyle\operatorname{div}(T_{\epsilon})$ $\displaystyle=2\langle\nabla
u,\nabla^{*}\nabla u\rangle+d\frac{W(u)}{\epsilon^{2}}+2\omega(\langle
iu,\nabla u\rangle,\cdot)-2\epsilon^{2}\omega(d^{*}\omega,\cdot)$
$\displaystyle=-2\langle\nabla
u,\dot{u}\rangle-2\epsilon^{2}d\alpha(\cdot,\dot{\alpha}),$

where the second equality follows from (6.1). We can now rewrite the term
$\langle dh,de_{t}\rangle$ in our computation of $\Phi_{h}^{\prime}(t)$ as

$\langle dh,de_{t}\rangle=\langle
dh,\operatorname{div}(T_{\epsilon})+2\operatorname{div}(\nabla u^{*}\nabla
u+\epsilon^{2}d\alpha^{*}d\alpha)\rangle,$

and apply the formula for $\operatorname{div}(T_{\epsilon})$ to see that

$\displaystyle\Phi_{h}^{\prime}(t)$ $\displaystyle=2\int_{M}\langle
dh,\operatorname{div}(\nabla u^{*}\nabla
u+\epsilon^{2}d\alpha^{*}d\alpha)\rangle$
$\displaystyle\quad-2\int_{M}h(|\dot{u}|^{2}+\epsilon^{2}|\dot{\alpha}|^{2})-4\int_{M}(\langle\nabla_{dh}u,\dot{u}\rangle+\epsilon^{2}d\alpha(dh,\dot{\alpha}))$
$\displaystyle=-2\int\langle D^{2}h,\nabla u^{*}\nabla
u+\epsilon^{2}d\alpha^{*}d\alpha\rangle$
$\displaystyle\quad-2\int_{M}(h|\dot{u}+h^{-1}\nabla_{dh}u|^{2}+\epsilon^{2}h|\dot{\alpha}+h^{-1}\iota_{dh}d\alpha|^{2})$
$\displaystyle\quad+2\int_{M}h^{-1}(|\nabla_{dh}u|^{2}+\epsilon^{2}|\iota_{dh}d\alpha|^{2})$
$\displaystyle\leq-2\int\langle D^{2}h,\nabla u^{*}\nabla
u+\epsilon^{2}d\alpha^{*}d\alpha\rangle+2\int_{M}h^{-1}(|\nabla_{dh}u|^{2}+\epsilon^{2}|\iota_{dh}d\alpha|^{2}).$

Now, setting

$P_{t}:=\nabla u^{*}\nabla u+\epsilon^{2}d\alpha^{*}d\alpha,$

so that the stress-energy tensor $T_{\epsilon}(u,\nabla)$ becomes simply
$e_{\epsilon}(u,\nabla)g-2P_{t}$, we can rewrite the preceding inequality as

(6.13) $\Phi_{h}^{\prime}(t)\leq-2\int_{M}\langle P_{t},D^{2}h-h^{-1}dh\otimes
dh\rangle.$

On the other hand, by Hamilton’s _matrix Harnack estimate_ for the heat
equation, see [29, p. 132], there exist constants $C(M)$ and $B(M)$ such that,
for $t\in[T-1,T)$,

$D^{2}h-\frac{dh\otimes
dh}{h}+\frac{1}{2(T-t)}hg\geq-C[(1+h\log(B/(T-t)^{n/2})]g.$

Applying this in (6.13), we see that for $t\in[T-1,T)$ the following
inequality holds:

$\Phi_{h}^{\prime}(t)\leq\int_{M}\Big{(}\frac{h}{T-t}+C+Ch\log(B/(T-t)^{n/2})\Big{)}\langle
P_{t},g\rangle.$

Now, recalling (6.12), observe that

$\displaystyle\langle P_{t},g\rangle$ $\displaystyle=|\nabla
u|^{2}+2\epsilon^{2}|d\alpha|^{2}$
$\displaystyle=e_{t}+\epsilon^{2}|d\alpha|^{2}-\frac{(1-|u|^{2})^{2}}{4\epsilon^{2}}$
$\displaystyle=e_{t}+\xi_{t}\Big{(}\epsilon|d\alpha|+\frac{1}{2\epsilon}(1-|u|^{2})\Big{)}$
$\displaystyle\leq(1+Ce^{Ct}\epsilon^{\frac{2}{n-1}})e_{t}+Ce^{Ct}\sqrt{e_{t}}$

for $t\geq 1$. In particular, setting $\alpha_{n}:=\frac{2}{n-1}$, for
$T\in[2,3]$ and $t\in[T-1,T)$ it then follows that

$\displaystyle\Phi_{h}^{\prime}(t)$
$\displaystyle\leq\frac{1+C\epsilon^{\alpha_{n}}}{T-t}\Phi_{h}(t)+\frac{C}{T-t}\int_{M}h\sqrt{e_{t}}+C\int_{M}e_{t}+C\log(B/(T-t)^{n/2})\Phi_{h}(t)$
$\displaystyle\leq\frac{1+C_{2}\epsilon^{\alpha_{n}}}{T-t}\Phi_{h}(t)+\frac{C_{2}}{T-t}\Phi_{h}(t)^{1/2}+C_{2}+C_{2}\log(B/(T-t)^{n/2})\Phi_{h}(t)$

for some $C_{2}(M,\Lambda)$, where we also used the trivial inequality
$\langle P_{t},g\rangle\leq 2e_{t}$. Thus, setting

$\Psi_{h}(t):=(T-t)^{1+C_{2}\epsilon^{\alpha_{n}}}e^{\zeta(t)}\Phi_{h}(t),$

where $|\zeta(t)|\leq C(M,\Lambda)$ is the bounded function on $[T-1,T)$ given
by

$\zeta(t):=-\int_{1}^{t}C_{2}\log(B/(T-s)^{n/2})\,ds,$

we see that

$\Psi_{h}^{\prime}(t)\leq C(T-t)^{-1/2}\Psi_{h}(t)^{1/2}+C$

for $t\in[T-1,T)\subseteq[1,3)$. From this differential inequality, we can
conclude that

(6.14) $\Psi_{h}(t)\leq C(M,\Lambda)(\Psi_{h}(T-1)+1),$

for any $t\in[T-1,T)\subseteq[1,3)$.

Specializing, fix $T\in[2,3]$ and $x_{0}\in M$, and let

$h(t,x)=h_{T,x_{0}}(t,x):=K(T-t,x,x_{0}),$

where $K$ is the heat kernel on $M$. Then, for $t\in[T-1,T)$, the inequality
in (6.14) leads to an estimate of the form

$\displaystyle(T-t)^{1+C\epsilon^{\alpha_{n}}}$
$\displaystyle\int_{M}K(T-t,x,x_{0})e_{\epsilon}(u_{t},\nabla_{t})\,dx$
$\displaystyle\leq C\int_{M}K(1,x,x_{0})e_{\epsilon}(u_{T-1},\nabla_{T-1})+C$
$\displaystyle\leq CE_{\epsilon}(u_{T-1},\nabla_{T-1})+C$ $\displaystyle\leq
C(M,\Lambda).$

In particular, taking $t:=2$ and $T:=2+\delta^{2}$ for $\delta\in(0,1]$, we
see that

(6.15)
$\delta^{2+2C\epsilon^{\alpha_{n}}}\int_{M}K(\delta^{2},x,x_{0})e_{\epsilon}(u_{2},\nabla_{2})\,dx\leq
C(M,\Lambda).$

Since

$\inf_{\epsilon\in(0,1]}\epsilon^{2C\epsilon^{\alpha_{n}}}=c(M,\Lambda)>0,$

it follows that

(6.16) $\max_{\epsilon\leq\delta\leq
1}\Big{(}\delta^{2}\int_{M}K(\delta^{2},x,x_{0})e_{\epsilon}(u_{2},\nabla_{2})\,dx\Big{)}\leq
C(M,\Lambda).$

Finally, using again Lemma 6.2, it follows that

$\delta^{2-n}\int_{B_{\delta}(x_{0})}e_{\epsilon}(u_{2},\nabla_{2})\leq
C(M,\Lambda)\quad\text{for }\epsilon\leq\delta\leq 1.$

Thus, we have arrived at the following bound.

######  Proposition 6.3.

If $(u_{t},\nabla_{t})$ is a solution of the gradient flow (6.1) for
$E_{\epsilon}$ with initial energy bound
$E_{\epsilon}(u_{0},\nabla_{0})\leq\Lambda$, then at time $2$ the pair
$(u_{2},\nabla_{2})$ satisfies

(6.17) $\int_{B_{r}(x_{0})}e_{\epsilon}(u_{2},\nabla_{2})\leq
C(M,\Lambda)r^{n-2},$

for all $r\in[\epsilon,1]$ and $\ x_{0}\in M$.

Since $(u_{2},\nabla_{2})$ depends continuously on the initial couple
$(u_{0},\nabla_{0})$, this provides in particular the regularization that we
needed in the previous section.

######  Remark 6.4.

Note that, in analogy with the monotonicity formula for critical couples, if
we just used the trivial bound $\langle P_{t},g\rangle\leq 2e_{t}$ we would
have obtained

$\Phi_{h}^{\prime}(t)\leq\frac{2}{T-t}\Phi_{h}(t)+C+C\log(B/(T-t)^{n/2})\Phi_{h}(t),$

leading to

$(T-t)^{2}\int_{M}h_{T,x_{0}}(t,\cdot)e_{t}\leq C(M,\Lambda)$

and hence a non-sharp bound $C\delta^{n-4}$ for the energy of
$(u_{2},\nabla_{2})$ on a ball $B_{\delta}(x_{0})$. This would have sufficed
for our present purposes (of ruling out concentration of mass in the min-max
families) only when $n>4$.

###  6.3. Long-time existence of the gradient flow

In this last part we show long-term existence, uniqueness and continuous
dependence on initial conditions for the gradient flow of $E_{\epsilon}$, on
the trivial line bundle. To do so, it is convenient to pass to the Coulomb
gauge. Namely, given a smooth couple $(u,\alpha)$, we can always find a change
of gauge

(6.18) $(v,\beta)=(e^{i\theta}u,\alpha+d\theta)\quad\text{with
$d^{*}\beta=0$}.$

Indeed, it is enough to take a solution $\theta:M\to\mathbb{R}$ of
$d^{*}\alpha+d^{*}d\theta=0$, i.e., $\Delta_{H}\theta=-d^{*}\alpha$. The
solution is unique once we impose $\int_{M}\theta=0$.

In the sequel, we denote
$Q:=-\Delta_{H}^{-1}d^{*}:\Omega^{1}(M)\to\Omega^{0}(M)$ the corresponding
operator, with values into mean-zero functions. By standard elliptic
regularity, this operator maps $H^{k}(M)$ continuously into $H^{k+1}(M)$, for
any $k\in\mathbb{N}$.

Given a smooth solution $(u_{t},\alpha_{t})$ to the gradient flow equations,
let $\theta_{t}=Q\alpha_{t}$. Omitting the time dependence and passing to the
Coulomb gauge as in (6.18) we get $\dot{\theta}=\epsilon^{-2}Q\langle
iu,\nabla u\rangle$. Thus, setting $\tilde{\nabla}:=d-i\beta=\nabla-id\theta$,
we obtain

$\displaystyle\dot{\beta}$ $\displaystyle=\dot{\alpha}+d\dot{\theta}$
$\displaystyle=-d^{*}d\alpha+\epsilon^{-2}(\langle iu,\nabla
u\rangle+dQ\langle iu,\nabla u\rangle)$
$\displaystyle=-\Delta_{H}\beta+\epsilon^{-2}(\langle
iv,\tilde{\nabla}v\rangle+dQ\langle iv,\tilde{\nabla}v\rangle),$

since by gauge invariance $d^{*}d\alpha=d^{*}d\beta=\Delta_{H}\beta$ and
$\langle iu,\nabla u\rangle=\langle iv,\tilde{\nabla}v\rangle$. Similarly,

$\displaystyle\dot{v}$
$\displaystyle=e^{i\theta}\dot{u}+ie^{i\theta}\dot{\theta}u$
$\displaystyle=-\tilde{\nabla}^{*}\tilde{\nabla}v+\frac{1}{2\epsilon^{2}}(1-|v|^{2})v+\epsilon^{-2}(Q\langle
iv,\tilde{\nabla}v\rangle)iv.$

Let $P:\Omega^{1}(M)\to\Omega^{1}(M)$ denote the Hodge projection on the co-
closed part of a one-form. Since $-dQ\lambda$ equals the exact part of
$\lambda$, we have $\lambda+dQ\lambda=P\lambda$ for any
$\lambda\in\Omega^{1}(M)$. Thus, expanding $\tilde{\nabla}^{*}\tilde{\nabla}$
in terms of $\beta$, the equations (6.1) give the new system

(6.19) $\left\\{\begin{aligned}
\dot{v}+d^{*}dv&=-2i\langle\beta,dv\rangle-|\beta|^{2}v+{\textstyle\frac{1}{2\epsilon^{2}}}(1-|v|^{2})v+\epsilon^{-2}(Q\langle
iv,dv-i\beta v\rangle)iv,\\\
\dot{\beta}+\Delta_{H}\beta&=\epsilon^{-2}P\langle iv,dv-i\beta
v\rangle.\end{aligned}\right.$

Conversely, given a couple $(u_{0},\alpha_{0})$ and setting
$\theta_{0}:=Q\alpha_{0}$, from a smooth solution $(v_{t},\beta_{t})$ of
(6.19) with initial condition $(e^{i\theta_{0}}u_{0},\alpha_{0}+d\theta_{0})$
one recovers a smooth solution $(u_{t},\alpha_{t})$ to the original system
(6.1), by letting $\theta=\theta_{t}$ solve
$\dot{\theta}_{t}=\epsilon^{-2}Q\langle iv_{t},(d-i\beta_{t})v_{t}\rangle$,
and setting $(u,\alpha):=(e^{-i\theta}v,\beta-d\theta)$.

Thus, we reduce ourselves to establishing the desired long-term existence,
uniqueness and continuous dependence for (6.19). We will use the following
classical fact from the theory of linear parabolic equations.

######  Lemma 6.5.

Given $f_{t}\in\Omega^{\ell}(M)$ smooth on $[0,T]\times M$, with $0<T\leq 1$,
the (unique) solution $w_{t}$ to $\partial_{t}w_{t}+\Delta_{H}w_{t}=f_{t}$
with initial condition $w_{0}=0$ satisfies

$\|w\|_{C^{0}([0,T],H^{k+1}(M))}\leq
C(k,\ell,M)\|f\|_{L^{2}([0,T],H^{k}(M))},$

where the norms are shorthand for $\max_{t\in[0,T]}\|w_{t}\|_{H^{k+1}(M)}$ and
$(\int_{0}^{T}\|f_{t}\|_{H^{k}(M)}^{2}\,dt)^{1/2}$.

As a consequence, we get a well-defined operator

$T_{\ell,k}:L^{2}([0,T],H^{k}(M))\to C^{0}([0,T],H^{k+1}(M))$

mapping $f$ to $w$.

Using this lemma, short-time existence and uniqueness easily follow using the
Banach fixed-point theorem. Namely, fix an integer $k>\frac{n}{2}$ and, given
a smooth initial condition $(v_{0},\beta_{0})$, let $w^{0}$ denote the
constant couple $w^{0}_{t}=(v_{0},\beta_{0})$. For $R>0$, the subset $S$ of

$\displaystyle Y_{T}:=C^{0}([0,T],H^{k+1}(M)\times H^{k}(M)),$

given by the couples $w_{t}$ with initial value $w_{0}=(v_{0},\beta_{0})$ and
$\|w-w^{0}\|_{Y_{T}}\leq R$, forms a complete metric space with the distance
induced by $Y_{T}$. To any $w=(v,\beta)\in S$ we can associate the solution
$F(w)=(v^{\prime},\beta^{\prime})$ of

$\displaystyle\left\\{\begin{aligned}
\dot{v^{\prime}}+d^{*}dv^{\prime}&=-2i\langle\beta,dv\rangle-|\beta|^{2}v+{\textstyle\frac{1}{2\epsilon^{2}}}(1-|v|^{2})v+\epsilon^{-2}(Q\langle
iv,dv-i\beta v\rangle)iv,\\\
\dot{\beta^{\prime}}+\Delta_{H}\beta^{\prime}&=\epsilon^{-2}P\langle
iv,dv-i\beta v\rangle.\end{aligned}\right.$

Denoting $G(w_{t})$ and $H(w_{t})$ the right-hand sides of the two equations,
note that they belong to $C^{0}([0,T],H^{k}(M))$, since $H^{k}(M)$ is an
algebra and $P$ and $Q$ map $H^{k}(M)$ into itself. Hence, $F(w)\in Y_{T}$ is
well-defined. For the same reason, letting
$R^{\prime}:=R+\|v_{0}\|_{H^{k+1}(M)}+\|\beta_{0}\|_{H^{k}(M)}$, note that for
a fixed $t\in[0,T]$ we have

$\|G(w_{t}^{1})-G(w_{t}^{2})\|_{H^{k}(M)}\leq
C(M,R^{\prime})\|w_{t}^{1}-w_{t}^{2}\|_{H^{k+1}(M)\times H^{k}(M)}$

and similarly

$\|H(w_{t}^{1})-H(w_{t}^{2})\|_{H^{k}(M)}\leq
C(M,R^{\prime})\|w_{t}^{1}-w_{t}^{2}\|_{H^{k+1}(M)\times H^{k}(M)},$

whenever $w^{1},w^{2}\in S$. As a consequence, Lemma 6.5 gives

$\|F(w^{1})-F(w^{2})\|_{Y_{T}}\leq
C(M,R^{\prime})\sqrt{T}\|w^{1}-w^{2}\|_{Y_{T}}.$

Hence, for $T$ small enough, we have
$\|F(w^{1})-F(w^{2})\|_{Y_{T}}\leq\frac{1}{2}\|w^{1}-w^{2}\|_{Y_{T}}$ and, by
continuity, $\|F(w^{0})-w^{0}\|_{Y_{T}}\leq R/2$; in particular,

$\|F(w)-w^{0}\|_{Y_{T}}\leq\|F(w)-F(w^{0})\|_{Y_{T}}+\|F(w^{0})-w^{0}\|_{Y_{T}}\leq
R$

for $w\in S$, and thus $F(w)\in S$ as well. The Banach fixed-point theorem
applies and gives a unique $w\in S$ with $F(w)=w$, as desired. Since $R$ was
arbitrary, this also establishes uniqueness in this regularity class.

Let $[0,\bar{T})$ be the maximal time of existence in the same class. From
standard $L^{2}$ regularity theory for linear parabolic equations, it then
follows that the solution $(v,\beta)$ is smooth on $[0,\bar{T})\times M$.

We shall now prove long-time existence of the flow. Assume by contradiction
that $\bar{T}<\infty$. As we already saw above, the corresponding solution
$(u,\alpha)$ to the original system (6.1) satisfies

$\sup_{[0,\bar{T})\times M}|d\alpha|<\infty.$

In a similar fashion, we can derive a bound for $|\nabla u|$. Indeed, as in
[50, Section 3], we have the Bochner identity

$\displaystyle-(\partial_{t}+dd^{*})\frac{1}{2}|\nabla
u|^{2}=|\nabla^{2}u|^{2}+\frac{3|u|^{2}-1}{2\epsilon^{2}}|\nabla
u|^{2}-2\langle\omega,\psi(u,\nabla)\rangle+\mathcal{R}_{1}(\nabla u,\nabla
u)$

and, in particular, using the bound $|\psi(u,\nabla)|\leq|\nabla u|^{2}$, we
easily deduce the weak subequation

$\displaystyle-(\partial_{t}+d^{*}d)|\nabla u|$
$\displaystyle\geq\frac{3|u|^{2}-1}{2\epsilon^{2}}|\nabla u|-2|\omega||\nabla
u|-C(M)|\nabla u|.$

Recalling that

$-(\partial_{t}+d^{*}d)\frac{1-|u|^{2}}{\epsilon}=\frac{|u|^{2}}{\epsilon^{2}}\frac{1-|u|^{2}}{\epsilon}-\frac{2}{\epsilon}|\nabla
u|^{2},$

we obtain for the difference $w:=|\nabla u|-\frac{1-|u|^{2}}{\epsilon}$ that

$\displaystyle-(\partial_{t}+d^{*}d)w\geq\frac{|u|^{2}}{\epsilon^{2}}w+|\nabla
u|\Big{(}\frac{2}{\epsilon}|\nabla
u|-\frac{1-|u|^{2}}{2\epsilon^{2}}-2|\omega|-C(M)\Big{)}.$

For any $0<\tau<\bar{T}$, if $w$ attains a positive maximum on $[0,\tau]\times
M$ at some point $(t,x)$ with $t>0$, it then follows that here

$\displaystyle\frac{2}{\epsilon}|\nabla
u|\leq\frac{1-|u|^{2}}{2\epsilon^{2}}+2|\omega|+C(M).$

Hence,

$|\nabla u|\leq\frac{1}{\epsilon}+\sup_{[0,\bar{T})\times
M}w\leq\frac{2}{\epsilon}+\epsilon\sup_{[0,\bar{T})\times
M}|\omega|+\frac{\epsilon}{2}C(M)+\|\nabla_{0}u_{0}\|_{L^{\infty}(M)}$

on all of $[0,\bar{T})\times M$. By gauge invariance, we then get

$\displaystyle\sup_{[0,\bar{T})\times
M}|d\beta|<\infty\quad\text{and}\quad\sup_{[0,\bar{T})\times M}|dv-i\beta
v|<\infty.$

In particular, the co-exact part of $\beta_{t}$ is also bounded. From (6.1) it
follows that

$\displaystyle\int_{M}(|\dot{u}_{t}|^{2}+\epsilon^{2}|\dot{\alpha}_{t}|^{2})=-\frac{1}{2}\frac{d}{dt}E_{\epsilon}(u_{t},\alpha_{t}),$

from which we deduce the bound
$\int_{0}^{\bar{T}}\int_{M}|\dot{\alpha}|^{2}<\infty$ just by integrating the
above expression. In particular, $\dot{\alpha}\in
L^{1}([0,\bar{T}],L^{2}(M))$, giving $\alpha\in C^{0}([0,\bar{T}],L^{2}(M))$.
Thus, the harmonic part $\alpha^{h}_{t}$ in the Hodge decomposition of
$\alpha_{t}$ stays bounded. Since $\beta_{t}^{h}=\alpha_{t}^{h}$ and $\beta$
has no exact part, this implies that

$\displaystyle\sup_{[0,\bar{T})\times M}|\beta|<\infty.$

Also, note that $|v|=|u|\leq 1$ as a simple application of the maximum
principle to the equation satisfied by $|u|^{2}$, provided $|u_{0}|\leq 1$,
implying

$\sup_{[0,\bar{T})\times M}|dv|<\infty.$

From $L^{p}$ regularity theory (see, e.g., [59]), it follows that $v,\beta\in
L^{p}([0,\bar{T}],W^{k,p}(M))$ for all $k\in\mathbb{N}$, $1<p<\infty$ and,
hence, $v$ and $\beta$ extend smoothly to $[0,\bar{T}]\times M$. Since we can
extend the solution past $\bar{T}$, we arrive at a contradiction. This shows
that $\bar{T}=\infty$. Finally, continuous dependence (in the smooth topology)
on the initial condition for the system (6.1) follows from the same property
for (6.19).

## References

  * [1] G. Alberti  “Variational models for phase transitions, an approach via $\Gamma$-convergence”  In _Calculus of variations and partial differential equations (Pisa, 1996)_ Springer, Berlin, 2000, pp. 95–114
  * [2] Giovanni Alberti, Sisto Baldo and Giandomenico Orlandi  “Functions with prescribed singularities”  In _J. Eur. Math. Soc. (JEMS)_ 5.3, 2003, pp. 275–311  DOI: 10.1007/s10097-003-0053-5
  * [3] Giovanni Alberti, Sisto Baldo and Giandomenico Orlandi  “Variational convergence for functionals of Ginzburg–Landau type”  In _Indiana Univ. Math. J._ 54.5, 2005, pp. 1411–1472  DOI: 10.1512/iumj.2005.54.2601
  * [4] Roberto Alicandro and Marcello Ponsiglione  “Ginzburg–Landau functionals and renormalized energy: a revised $\Gamma$-convergence approach”  In _J. Funct. Anal._ 266.8, 2014, pp. 4890–4907  DOI: 10.1016/j.jfa.2014.01.024
  * [5] Frederick Justin Almgren  “The homotopy groups of the integral cycle groups”  In _Topology_ 1, 1962, pp. 257–299  DOI: 10.1016/0040-9383(62)90016-2
  * [6] Frederick Justin Almgren  “The theory of varifolds (Mimeographed notes)”, 1965
  * [7] F. Bethuel, H. Brezis and F. Hélein  “Ginzburg–Landau vortices” 13, Progress in Nonlinear Differential Equations and their Applications  Birkhäuser Boston, Inc., Boston, MA, 1994  DOI: 10.1007/978-1-4612-0287-5
  * [8] F. Bethuel, H. Brezis and G. Orlandi  “Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions”  In _J. Funct. Anal._ 186.2, 2001, pp. 432–520  DOI: 10.1006/jfan.2001.3791
  * [9] F. Bethuel and T. Rivière  “Vorticité dans les modèles de Ginzburg–Landau pour la supraconductivité (Exp. No. XVI)”  In _Séminaire sur les Équations aux Dérivées Partielles, 1993–1994_ École Polytech., Palaiseau, 1994
  * [10] F. Bethuel and T. Rivière  “Vortices for a variational problem related to superconductivity”  In _Ann. Inst. H. Poincaré Anal. Non Linéaire_ 12.3, 1995, pp. 243–303  DOI: 10.1016/S0294-1449(16)30157-3
  * [11] Andrea Braides  “$\Gamma$-convergence for beginners” 22, Oxford Lecture Series in Mathematics and its Applications  Oxford University Press, Oxford, 2002  DOI: 10.1093/acprof:oso/9780198507840.001.0001
  * [12] Haïm Brezis, Jean-Michel Coron and Elliott H. Lieb  “Harmonic maps with defects”  In _Comm. Math. Phys._ 107.4, 1986, pp. 649–705  URL: http://projecteuclid.org/euclid.cmp/1104116234
  * [13] Giacomo Canevari and Giandomenico Orlandi  “Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces”  In _Calc. Var. Partial Differential Equations_ 58.2, art. 72, 2019  DOI: 10.1007/s00526-019-1501-8
  * [14] Giacomo Canevari and Giandomenico Orlandi  “Topological singular set of vector-valued maps, II: $\Gamma$-convergence for Ginzburg–Landau type functionals”, 2020  arXiv:2003.01354 [math.AP]
  * [15] Isaac Chavel  “Eigenvalues in Riemannian geometry” 115, Pure and Applied Mathematics  Academic Press, Inc., Orlando, FL, 1984
  * [16] Da Rong Cheng  “Asymptotics for the Ginzburg–Landau equation on manifolds with boundary under homogeneous Neumann condition”  In _J. Funct. Anal._ 278.4, 2020, pp. 108364+93  DOI: 10.1016/j.jfa.2019.108364
  * [17] Otis Chodosh and Christos Mantoulidis  “Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates”  In _Ann. of Math. (2)_ 191.1, 2020, pp. 213–328  DOI: 10.4007/annals.2020.191.1.4
  * [18] Tobias H. Colding and Camillo De Lellis  “The min-max construction of minimal surfaces”  In _Surveys in differential geometry, Vol. VIII_ Int. Press, Somerville, MA, 2003, pp. 75–107
  * [19] Gianni Dal Maso  “An introduction to $\Gamma$-convergence” 8, Progress in Nonlinear Differential Equations and their Applications  Birkhäuser Boston, Inc., Boston, MA, 1993  DOI: 10.1007/978-1-4612-0327-8
  * [20] Ennio De Giorgi and Tullio Franzoni  “Su un tipo di convergenza variazionale”  In _Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8)_ 58.6, 1975, pp. 842–850
  * [21] Françoise Demengel  “Une caractérisation des applications de $W^{1,p}(B^{N},S^{1})$ qui peuvent être approchées par des fonctions régulières”  In _C. R. Acad. Sci. Paris Sér. I Math._ 310.7, 1990, pp. 553–557
  * [22] Akashdeep Dey  “A comparison of the Almgren–Pitts and the Allen–Cahn min-max theory”, 2020  arXiv:2004.05120 [math.DG]
  * [23] Herbert Federer  “Geometric Measure Theory”  Springer Berlin Heidelberg, 2014  URL: https://books.google.co.uk/books?id=jld-BgAAQBAJ
  * [24] Pedro Gaspar and Marco A. M. Guaraco  “The Allen–Cahn equation on closed manifolds”  In _Calc. Var. Partial Differential Equations_ 57.4, art. 101, 2018  DOI: 10.1007/s00526-018-1379-x
  * [25] Nassif Ghoussoub  “Duality and perturbation methods in critical point theory” 107, Cambridge Tracts in Mathematics  Cambridge University Press, Cambridge, 1993  DOI: 10.1017/CBO9780511551703
  * [26] Phillip Griffiths and Joseph Harris  “Principles of algebraic geometry”  John Wiley & Sons, Inc., New York, 1994  DOI: 10.1002/9781118032527
  * [27] Marco A. M. Guaraco  “Min-max for phase transitions and the existence of embedded minimal hypersurfaces”  In _J. Differential Geom._ 108.1, 2018, pp. 91–133  DOI: 10.4310/jdg/1513998031
  * [28] Morton E. Gurtin  “Some results and conjectures in the gradient theory of phase transitions”  In _Metastability and incompletely posed problems (Minneapolis, Minn., 1985)_ 3, IMA Vol. Math. Appl.  Springer, New York, 1987, pp. 135–146
  * [29] Richard S. Hamilton  “Monotonicity formulas for parabolic flows on manifolds”  In _Comm. Anal. Geom._ 1.1, 1993, pp. 127–137  DOI: 10.4310/CAG.1993.v1.n1.a7
  * [30] Zheng-Chao Han and Yan Yan Li  “Degenerate elliptic systems and applications to Ginzburg–Landau type equations, I”  In _Calc. Var. Partial Differential Equations_ 4.2, 1996, pp. 171–202  DOI: 10.1007/s005260050034
  * [31] R. Hardt, D. Kinderlehrer and Fanghua Lin  “Stable defects of minimizers of constrained variational principles”  In _Ann. Inst. H. Poincaré Anal. Non Linéaire_ 5.4, 1988, pp. 297–322
  * [32] Allen Hatcher  “Algebraic topology”  Cambridge University Press, Cambridge, 2002
  * [33] Min-Chun Hong  “Asymptotic behavior for minimizers of a Ginzburg–Landau-type functional in higher dimensions associated with $n$-harmonic maps”  In _Adv. Differential Equations_ 1.4, 1996, pp. 611–634
  * [34] Min-Chun Hong, Jürgen Jost and Michael Struwe  “Asymptotic limits of a Ginzburg–Landau type functional”  In _Geometric analysis and the calculus of variations_ Int. Press, Cambridge, MA, 1996, pp. 99–123
  * [35] Gerhard Huisken  “Asymptotic behavior for singularities of the mean curvature flow”  In _J. Differential Geom._ 31, 1990, pp. 285–299
  * [36] John E. Hutchinson and Yoshihiro Tonegawa  “Convergence of phase interfaces in the van der Waals–Cahn–Hilliard theory”  In _Calc. Var. Partial Differential Equations_ 10.1, 2000, pp. 49–84  DOI: 10.1007/PL00013453
  * [37] Tom Ilmanen  “Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature”  In _J. Differential Geom._ 38.2, 1993, pp. 417–461
  * [38] Arthur Jaffe and Clifford H. Taubes  “Vortices and monopoles” 2, Progress in Physics  Birkhäuser, Boston, Mass., 1980
  * [39] Robert L. Jerrard  “Lower bounds for generalized Ginzburg–Landau functionals”  In _SIAM J. Math. Anal._ 30.4, 1999, pp. 721–746  DOI: 10.1137/S0036141097300581
  * [40] Robert L. Jerrard and Halil Mete Soner  “The Jacobian and the Ginzburg–Landau energy”  In _Calc. Var. Partial Differential Equations_ 14.2, 2002, pp. 151–191  DOI: 10.1007/s005260100093
  * [41] Robert L. Jerrard and Halil Mete Soner  “Functions of bounded higher variation”  In _Indiana Univ. Math. J._ 51.3, 2002, pp. 645–677  DOI: 10.1512/iumj.2002.51.2229
  * [42] Antoni A. Kosinski  “Differential manifolds” 138, Pure and Applied Mathematics  Academic Press, Inc., Boston, MA, 1993
  * [43] Fanghua Lin and Tristan Rivière  “Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents”  In _J. Eur. Math. Soc. (JEMS)_ 1.3, 1999, pp. 237–311  DOI: 10.1007/s100970050008
  * [44] Fanghua Lin and Tristan Rivière  “A quantization property for static Ginzburg–Landau vortices”  In _Comm. Pure Appl. Math._ 54.2, 2001, pp. 206–228
  * [45] Yevgeny Liokumovich, Fernando C. Marques and André Neves  “Weyl law for the volume spectrum”  In _Ann. of Math. (2)_ 187.3, 2018, pp. 933–961  DOI: 10.4007/annals.2018.187.3.7
  * [46] Fernando C. Marques and André Neves  “Min-max theory and the Willmore conjecture”  In _Ann. of Math. (2)_ 179.2, 2014, pp. 683–782  DOI: 10.4007/annals.2014.179.2.6
  * [47] Luciano Modica  “The gradient theory of phase transitions and the minimal interface criterion”  In _Arch. Rational Mech. Anal._ 98.2, 1987, pp. 123–142  DOI: 10.1007/BF00251230
  * [48] Luciano Modica and Stefano Mortola  “Un esempio di $\Gamma^{-}$-convergenza”  In _Boll. Un. Mat. Ital. B (5)_ 14.1, 1977, pp. 285–299
  * [49] James R. Munkres  “Elementary differential topology (revised edition)” 54, Annals of Mathematics Studies  Princeton University Press, Princeton, N.J., 1961
  * [50] Alessandro Pigati and Daniel Stern  “Minimal submanifolds from the abelian Higgs model”  In _Inventiones mathematicae_ 223.3, 2021, pp. 1027–1095  DOI: 10.1007/s00222-020-01000-6
  * [51] Jon T. Pitts  “Existence and regularity of minimal surfaces on Riemannian manifolds” 27, Mathematical Notes  Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981
  * [52] Tristan Rivière  “Line vortices in the ${\rm U}(1)$-Higgs model”  In _ESAIM Contrôle Optim. Calc. Var._ 1, 1995/96, pp. 77–167  DOI: 10.1051/cocv:1996103
  * [53] Tristan Rivière  “Asymptotic analysis for the Ginzburg–Landau equations”  In _Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)_ 2.3, 1999, pp. 537–575
  * [54] Tristan Rivière  “Towards Jaffe and Taubes conjectures in the strongly repulsive limit”  In _Manuscripta Math._ 108.2, 2002, pp. 217–273  DOI: 10.1007/s002290200266
  * [55] Etienne Sandier  “Lower bounds for the energy of unit vector fields and applications”  In _J. Funct. Anal._ 152.2, 1998, pp. 379–403  DOI: 10.1006/jfan.1997.3170
  * [56] Etienne Sandier  “Ginzburg–Landau minimizers from $\mathbb{R}^{n+1}$ to $\mathbb{R}^{n}$ and minimal connections”  In _Indiana Univ. Math. J._ 50.4, 2001, pp. 1807–1844  DOI: 10.1512/iumj.2001.50.1751
  * [57] Etienne Sandier and Sylvia Serfaty  “A product-estimate for Ginzburg–Landau and corollaries”  In _J. Funct. Anal._ 211.1, 2004, pp. 219–244  DOI: 10.1016/S0022-1236(03)00199-X
  * [58] Etienne Sandier and Sylvia Serfaty  “Vortices in the magnetic Ginzburg–Landau model” 70, Progress in Nonlinear Differential Equations and their Applications  Birkhäuser Boston, Inc., Boston, MA, 2007
  * [59] Wilhelm Schlag  “Schauder and $L^{p}$ estimates for parabolic systems via Campanato spaces”  In _Comm. Partial Differential Equations_ 21.7–8, 1996, pp. 1141–1175  DOI: 10.1080/03605309608821221
  * [60] Sylvia Serfaty and Ian Tice  “Lorentz space estimates for the Ginzburg–Landau energy”  In _J. Funct. Anal._ 254.3, 2008, pp. 773–825  DOI: 10.1016/j.jfa.2007.11.010
  * [61] Leon Simon  “Lectures on geometric measure theory” 3, Proceedings of the Centre for Mathematical Analysis  Australian National University, Centre for Mathematical Analysis, Canberra, 1983
  * [62] Daniel Stern  “Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds”  In _J. Diff. Geom. (to appear)_ , 2019
  * [63] Peter Sternberg  “The effect of a singular perturbation on nonconvex variational problems”  In _Arch. Rational Mech. Anal._ 101.3, 1988, pp. 209–260  DOI: 10.1007/BF00253122
  * [64] Michael Struwe  “On the asymptotic behavior of minimizers of the Ginzburg–Landau model in $2$ dimensions”  In _Differential Integral Equations_ 7.5–6, 1994, pp. 1613–1624
  * [65] Clifford H. Taubes  “Arbitrary $N$-vortex solutions to the first order Ginzburg–Landau equations”  In _Comm. Math. Phys._ 72.3, 1980, pp. 277–292
  * [66] Clifford H. Taubes  “On the equivalence of the first and second order equations for gauge theories”  In _Comm. Math. Phys._ 75.3, 1980, pp. 207–227
  * [67] Yoshihiro Tonegawa and Neshan Wickramasekera  “Stable phase interfaces in the van der Waals–Cahn–Hilliard theory”  In _J. Reine Angew. Math._ 668, 2012, pp. 191–210  DOI: 10.1515/crelle.2011.134
  * [68] Brian White  “Infima of energy functionals in homotopy classes of mappings”  In _J. Differential Geom._ 23.2, 1986, pp. 127–142  URL: http://projecteuclid.org/euclid.jdg/1214440023


API

    application programming interface
BT

    behavior tree
DT

    digital twin
eBT

    extended behavior tree
FSM

    finite state machine
GUI

    graphical user interface
HTN

    hierarchical task network
MES

    manufacturing execution system
MG

    motion generator
BTMG

    behavior tree and motion generator
OWL

    Web Ontology Language
PDDL

    Planning Domain Definition Language
RDF

    Resource Description Framework
RL

    reinforcement learning
ROS

    Robot Operating System
WM

    world model
EE

    end effector

# Using Knowledge Representation and Task Planning for Robot-agnostic Skills
on the Example of Contact-Rich Wiping Tasks

Matthias Mayr1, Faseeh Ahmad1, Alexander Duerr1 and Volker Krueger1
1Department of Computer Science, Faculty of Engineering (LTH), Lund
University, SE 221 00 Lund, Sweden. E-mail<EMAIL_ADDRESS>
###### Abstract

The transition to agile manufacturing, Industry 4.0, and high-mix-low-volume
tasks require robot programming solutions that are flexible. However, most
deployed robot solutions are still statically programmed and use stiff
position control, which limit their usefulness.

In this paper, we show how a single robot skill that utilizes knowledge
representation, task planning, and automatic selection of skill
implementations based on the input parameters can be executed in different
contexts. We demonstrate how the skill-based control platform enables this
with contact-rich wiping tasks on different robot systems. To achieve that in
this case study, our approach needs to address different kinematics, gripper
types, vendors, and fundamentally different control interfaces. We conducted
the experiments with a mobile platform that has a Universal Robots UR5e 6
degree-of-freedom robot arm with position control and a 7 degree-of-freedom
KUKA iiwa with torque control.

##  I Introduction

The need for flexible and skill-based robot control systems in industrial
robot applications is becoming increasingly important. With the rise of
automation and the growing complexity of tasks, robots must be able to adapt
to changing environments and perform a variety of tasks without lengthy
configuration times. This requires a control system that can respond quickly
and accurately to changes in the environment and the task at hand. However,
most of the deployed systems are statically programmed and perform repetitive
tasks. This introduces the need for a flexible and skill-based robot control
system in industrial robot applications that provides flexibility and supports
agile manufacturing.

First platforms for skill-based systems such as ClaraTy [1] or LAAS [2], paved
the way. In the area of knowledge integration frameworks, the system built
around the Rosetta ontology [3, 4] and Knowrob [5, 6] created a solid
foundation. The latter addresses the service domain, while the former is
placed in the context of industrial robotics. To transfer skills from one
robot to another is nontrivial if robots of different types and vendors are
involved. For instance, [7] demonstrated skill transfer for contact-free
motions within a single family of robots that share the same vendor-supplied
high-level interface ABB RAPID. In that paper, the simulated experiments were
conducted with high-gain position control. This is, of course, still a
standard practice within many tasks that have little variation and high
certainty. However, the increasing number of contact-rich tasks that
manipulators should be able to solve [8] indicates a clear need for flexible
and compliant systems. Achieving compliant behaviors heavily depends on the
hardware and vendor, and there are no standard interfaces available for this
purpose. In this paper, we explore the question of how skills can be
transferred between collaborative robots, such as the _Universal Robot_ robots
and the _KUKA iiwa_ or the _Franka Emika Robot (Panda)_. To do this, we
develop a suitable representation of the available robot hardware, a knowledge
infrastructure to maintain this hardware knowledge, and a suitable compliant
controller that works across different types of robot arms. To validate and
verify our claims, we test our architecture on a whiteboard wiping task, as
shown in Fig. 1, using a _UR5e_ and a _KUKA iiwa_. The whiteboard wiping task
is an example of a typical contact-rich task that requires a compliant
controller. For this, the _UR5e_ has a force-torque sensor on the wrist, while
robots such as the _iiwa_ and the _Panda_ have torque sensors in their joints.

Figure 1: With the proposed abstractions and knowledge representation, several
different robot systems can execute the same skills while still using their
respective hardware such as grippers and controllers.

In summary, our paper makes the following contributions:

  1. 1.

We present a complete architecture that represents, plans, and executes skills
such that they can be transfered between robots.

  2. 2.

We demonstrate these ideas based on a whiteboard wiping task using a _UR5e_
and a _KUKA iiwa_ robot.

  3. 3.

All essential building blocks, including the impedance controller
implementations are available as open-source software.

  4. 4.

The code is written in a robot-agnostic way, making the results and insights
scalable to diverse collaborative robotic hardware and tasks.

##  II Related Work

###  II-A Robot Skill Systems

We can classify skill-based architectures based on their target domain.
Projects like CAST [9] and KnowRob [5, 6] are for service robotics and general
platforms are ClaraTy [1], LAAS [2] and SmartMDSD [10]. However, platforms
that address the specific needs of industrial robotics are the most relevant
for this work. CoStar [11] has shown to integrate multiple robot systems and
provides a graphical user interface (GUI) for the design and the monitoring of
skills. However, it lacks explicit knowledge representation and the
possibility to perform task planning. The use of ontologies to describe skills
in robotic applications has been explored in various areas, including medical
and surgical robotics [12, 13], perception ontologies [14] for robots, and
more [15]. Among these, [3, 4, 7] present a knowledge-based approach for
programming synchronized motions between robotic systems and human-robot
interactions that is particularly relevant for our work.

In [7], the authors demonstrate the effectiveness of their approach through
experiments that involve transformations of dual arm programs and the transfer
of skills between robot systems with different kinematics. Their approach uses
finite state machines to implement skill behavior and generates ABB RAPID code
that is specific to ABB robot controllers. Our work differs in that we use
behavior trees for skill behavior implementation and our implementation is not
limited to a particular robot controller. In addition, [7] was validated only
in simulation, and the tasks were not contact-rich.

###  II-B Robot Motions

In the literature, Movement Primitives (MPs) have been a reliable tool for
generating robot motions. Dynamic Movement Primitives (DMPs) [16, 17] and
Probabilistic Movement Primitives (ProMPs) [18] have been successfully used to
generate arm motions. Both DMPs and a motion representation based on behavior
trees and motion generators (BTMG) that will be introduced in Section IV-A use
attractor landscapes to reach a target location. DMPs are non-linear dynamical
systems that allow trajectory control, and, like our BTMGs, they have a non-
linear forcing term that enables the modification of the original trajectory.
However, in contrast to DMPs, the parameters of BTMGs can be specified
explicitly, giving us a better understanding of the robot motion and allowing
for a better intregration with knowledge representation.

Wiping Applications: In [19, 20, 21] the authors discuss research on robotic
manipulation of wiping tasks using artificial intelligence reasoning methods.
[19] proposes a classification of compliant manipulation tasks based on
symbolic effects, subcategorizes wiping tasks, and demonstrates how to
concretize actions based on the example of shards sweeping with a broom. [20]
investigates the reasoning and action execution problems involved in the
execution of wiping tasks and proposes a high-level abstraction representation
of wiping tasks to develop generalized action execution mechanisms. [21]
combines reasoning methods and compliant robotic manipulation to solve wiping
tasks by introducing a qualitative particle distribution model, an approach to
generate whole-body wiping motions based on effect-oriented policies, an
approach to assess wiping motion quality, and the semantic interpretation of
contact situations. Rather than achieving the best wiping skill, the focus of
our paper is on presenting a complete architecture that represents, plans, and
executes skills, and then using wiping as a challenging example to demonstrate
these ideas.

##  III Transferable Robot Skills

In this section we introduce the prerequisites that enable to transfer skills
between different robot systems. In addition to the skill platform SkiROS2 and
its skill model, the necessary aspects of knowledge presentation and task
planning are introduced. The control implementations and motion generation are
introduced in Section IV.

###  III-A SkiROS2

Figure 2: An outline of the SkiROS2 architecture. The world model stores the
knowledge about the relations, environment and the skills. The skill manager
loads and executes the skills. Dashed lines show control flows and solid lines
show information flows. Shaded blocks indicate possible multiple instances.

We utilize the skill-based robot control platform SkiROS2 [22] to implement
transferable robot skills. It is used in several research projects for motion
generation [23], robot coordination [8] and task-level planning with
reinforcement learning (RL) [24, 25, 26, 27]. The architecture is shown in
Fig. 2 and the main components are the skill manager and the world model. The
world model (WM) contains the knowledge about the world and represents it in a
Resource Description Framework (RDF) database. Ontologies such as the
Suggested Upper Merged Ontology (SUMO) describe the available concepts such as
objects or skills and their relations to each other. In addition to that, a
scene contains the concrete instances of the concepts that are defined in the
ontologies. Such instances can be the present robot systems, workstations or
objects that are currently known.

The skill manager is responsible for loading a set of specified skills from
the skill libraries. Each skill that is successfully loaded gets a semantic
description in the WM. This includes the parameters and pre-, hold- and post-
conditions. In addition to that, the skill manager provides services to start,
pause and stop skills and to manage their execution.

The task manager accepts the planning goals in Planning Domain Definition
Language (PDDL) [28, 22]. Such goals can come from a user or an external
entity such as a manufacturing execution system (MES). An example of such a
goal is to place an object at a specific location: (skiros:contain
skiros:Location-1 skiros:Product-1). The task manager automatically creates a
PDDL planning domain from the knowledge in the WM. The currently available
skills are added with their preconditions and effects, and the relevant
instances in the scene are automatically added to a planning problem. Then a
PDDL planner is called to find a sequence of skills that is guaranteed to be
optimal [29]. Finally, the plan is converted back into a behavior tree (BT),
automatically expanded with the relevant skill implementations and grounded
with the instances in the WM [22].

###  III-B Skill Model

We define a skill as a parametric procedure that changes the world from some
initial state to some new state [30]. It is shown in Fig. 3. A skill consists
of two main components: a skill description and the skill implementation. A
skill description contains the skill parameters and the necessary pre-, hold-
and post-conditions. These define what prerequisites need to exist to execute
a skill and state the expected effects of running it. In the task manager,
these conditions can be used for automatic task planning. Furthermore, the
parameters state the input and output of a skill. There are three different
types of parameters: 1) required, 2) optional and 3) inferred. An example of a
required parameter of a "pick" skill is to specify the robot arm, while an
optional parameter could be to slow down the execution for testing. The
inferred parameters are reasoned about when starting the skill. An example is
a "gripper" parameter for a pick skill that already has the arm as a required
parameter. With a precondition we can define a relation between both of them
and we can use the knowledge in the WM to specify the inferred parameter
automatically [22].

Figure 3: The conceptual model of a skill in SkiROS2. Pre-, and hold-
conditions ensure that the skill is only executed in the correct world state.
Post-conditions check if the desired changes have been achieved.

In addition to the skill description that describes the semantic actions of a
skill, the skill implementation is a concrete version for the execution. Each
skill implementation implements exactly one description with its parameters
and pre-, hold- and post-conditions. There are two complexities of skills:
Primitive skills that are semantically atomic and compound skills. Primitive
skills provide a Python interface for the initialization, start, execution and
stop of a skill. In contrast to that, compound skills allow to connect an
arbitrary number of primitive skills and compound skills in a BT [31, 22]. A
skill implementation is also allowed to change the skill description, such as
adding addition conditions or modifying existing conditions. This is important
to support multiple, specific skill implementations. Examples of this are
gripper actuation skills. A basic skill description can have a parameter
"gripper" of the WM type Element("rparts:GripperEffector") and a Boolean
parameter "opening state". The implementation of such a skill depends on the
actual gripper model and a specific implementation can set the parameter to a
specific subtype of rparts:GripperEffector, such as scalable:RobotiqGripper.

###  III-C Knowledge Representation and Task Planning

An important aspect is the separation of skills and tasks. Skills are written
in a parametric way to be used in various applications and different tasks.
This separation is achieved by explicitly storing knowledge in the WM. Since
the conditions and task-specific knowledge is not implicitly represented in
the skills, this allows for a modular and extensible design that easily allows
to add scenarios or robot systems at a later point. The WM stores this
knowledge in a semantic RDF database that utilizes the Web Ontology Language
(OWL). SkiROS2 introduces its own ontology, which contains the necessary
concepts and relations for skill modeling and reasoning. It is based on the
Core Ontology for Robotics and Automation (CORA) that is in the IEEE Standard
Ontologies for Robotics and Automation (IEEE Std
1872™-2015)color=green!40]ieee standard.

An example for a part of the semantic description of a robot arm is in Listing
1. The listing contains the relevant knowledge for the trajectory generation
and motion execution. This includes, but is not limited to properties such as
Robot Operating System (ROS) interfaces of a robot and the relevant aspects of
a motion planning configuration. In addition there are elements needed for
reasoning about relations such as the attached gripper.

Listing 1: An excerp from the semantic description of one of the robots used
in the case study. It includes the necessary knowledge to parameterize motion
skills and perform spatial reasoning.

⬇

scalable:Ur5-2 a scalable:Ur5, owl:NamedIndividual ;

rdfs:label "scalable:ur5"^^xsd:string ;

skiros:BaseFrameId "cora:Robot-1"^^xsd:string ;

skiros:CartesianGoalAction
"/cartesian_trajectory_generator/goal_action"^^xsd:string ;

skiros:OverlayMotionService
"/cartesian_trajectory_generator/overlay_motion"^^xsd:string ;

skiros:CartesianStiffnessTopic
"/cartesian_param_filter/stiffness_goal"^^xsd:string ;

skiros:CartesianWrenchTopic "/cartesian_param_filter/force_goal"^^xsd:string ;

skiros:CompliantController "cartesian_compliance_controller"^^xsd:string ;

skiros:JointConfigurationController "scaled_pos_traj_controller"^^xsd:string ;

skiros:DiscreteReasoner "AauSpatialReasoner"^^xsd:string ;

skiros:FrameId "scalable:Ur5-2"^^xsd:string ;

skiros:LinkedToFrameId "ur5e_base_link"^^xsd:string ;

skiros:MotionExe
"/scaled_pos_traj_controller/follow_joint_trajectory"^^xsd:string ;

skiros:MoveItGroup "manipulator"^^xsd:string ;

skiros:MoveItReferenceFrame "ur5e_base_link"^^xsd:string ;

skiros:MoveItTCPLink "ur5e_tcp_link"^^xsd:string ;

skiros:hasA scalable:WsgGripper-3 .

Tasks are executed by the skill manager and there are two ways to start them:
1) Select and parameterize a skill manually. For example in the GUI or from an
application programming interface (API) or 2) by providing a planning goal to
the task manager and executing the skill sequence. Tasks are concrete skill
(sequence) instances with a specific goal. Within a task, the skills can
utilize a blackboard to exchange information with their parameters.

##  IV Control and Motion Generation

###  IV-A Behavior Trees and Motion Generators

Behavior Trees (BT) are mathematical models used for plan representation and
execution. They have been shown to perform reliably in games [32], artificial
intelligence and robotics [33]. A BT is an acyclic graph defined with nodes
and edges defined in a typical parent-child relationship. The nodes are of two
types: 1) control flow and 2) execution. The control flow nodes are
responsible for directing the flow of the tree and are traditionally
classified on the basis of their operation as: 1) sequence, 2) selector, 3)
parallel and 4) decorator [31, 34]. In essence, a BT works by means of sending
a tick signal from the Root node. The signal then moves through the nodes
controlled by the control flow nodes. The return statements from the execution
nodes are _running_ , success or failure. In order to combine BTs with task-
level planning, [22] proposes extended BTs.

A motion generator (MG) [23] generates an arm motion using impedance
controllers to control the end effector (EE) in Cartesian space. A MG follows
a simple attractor landscape that uses a virtual spring to attract the EE to
the desired location, whereas virtual dampers allow safe motion by slowing
down the overall motion. In addition, it also allows for a deviation by
superimposing a motion on top of the original trajectory. For example, in [25]
we used a Cartesian linear trajectory superimposed by an Archimedes spiral to
solve the peg-in-hole task.

A Behavior Tree and Movement Generator (BTMG) combines the strengths of both
BT and MG to model robotic skills. A BTMG is a parameterized representation
that allows us to specify not only the structure of BT but also the properties
of the MG. In essence, the BTMG representation not only covers the aspects of
plan representation and execution, but also determines the arm motions for the
execution strategies [35].

###  IV-B Trajectory Generation

The concept of the MG has been introduced in [23], which consists of both a
trajectory generation method and a corresponding controller to execute the
generated trajectory on the manipulator. In the following section, we provide
a description of the trajectory generation process adopted for this project.

In many industrial settings, operations are performed in a Cartesian task
space rather than in the joint space of the robot. Therefore, we argue that
motion references are best generated and provided in the Cartesian end-
effector space. In this work, we adopt Cartesian linear
paths111https://github.com/matthias-mayr/cartesian_trajectory_generator to
generate the corresponding trajectories. This is achieved by selecting
appropriate acceleration profiles and setting the maximum Cartesian
translational and rotational velocities. Furthermore, we offer the option to
synchronize motions that have both translational and rotational components.
Specifically, the trajectory generator takes the new goal pose as input and
outputs end-effector pose references, which are then sent to the corresponding
controller.

Aside from specifying a new goal pose, it is also possible to overlay
additional motions on top of the reference pose. This capability has been
utilized in various studies such as [36, 24, 25], where an Archimedes spiral
is used to find a hole in an insertion procedure. In this work, we apply a
sine motion to improve the wiping performance of the robot.

###  IV-C Compliant Control

Manipulation tasks that have uncertainties need compliant control solutions to
provide the necessary flexibility. Such uncertainties can come from either the
task itself or inaccurate knowledge about the placement of objects. Another
source could be the placement of the robot, which becomes relevant when
working with a mobile robot. The robot systems used in the case study provide
different interfaces for control commands and therefore also have different
solutions for compliant control.

#### Cartesian Impedance Control

Robot systems such as the KUKA iiwa or the Franka Emika Robot offer an
interface to send commanded torques for each of the joints. This can be used
to perform Cartesian impedance control [37] which allows compliant control of
the end effector in task space.

Torque-controlled robots are typically gravity compensated. The rigid-body
dynamics of such a system in the joint space is described as
$q\in\mathbb{R}^{n}$ color=green!40]springer 2016:

$M(q)\ddot{q}+C(q,\dot{q})\dot{q}=\tau_{\mathrm{c}}+\tau^{\mathrm{ext}}$ (1)

where $M(q)\in\mathbb{R}^{n\times n}$ is the generalized inertia matrix,
$C(q,\dot{q})\in\mathbb{R}^{n\times n}$ captures the effects of Coriolis and
centripetal forces, $\tau_{\mathrm{c}}\in\mathbb{R}^{n}$ represents the input
torques, and $\tau^{\mathrm{ext}}\in\mathbb{R}^{n}$ represents the external
torques, with $n$ being the number of joints of the robot.

Moreover, the torque signal commanded by this controller to the robot,
$\tau_{\mathrm{c}}$ in Equation (1), is composed of the superposition of three
joint-torque signals:

$\tau_{\mathrm{c}}=\tau_{\mathrm{c}}^{\mathrm{ca}}+\tau_{\mathrm{c}}^{\mathrm{ns}}+\tau_{\mathrm{c}}^{\mathrm{ext}}$
(2)

where

$\tau_{\mathrm{c}}^{\mathrm{ca}}$ is the torque commanded to achieve a
Cartesian impedance behavior with respect to a Cartesian pose reference in the
$m$-dimensional task space, $\xi^{\mathrm{D}}\in\mathbb{R}^{m}$, in the frame
of the end effector of the robot:

$\tau_{\mathrm{c}}^{\mathrm{ca}}=J^{\mathrm{T}}(q)\left[-K^{\mathrm{ca}}\Delta\xi-D^{\mathrm{ca}}J(q)\dot{q}\right]$
(3)

with $J(q)\in\mathbb{R}^{m\times n}$ being the Jacobian relative to the end-
effector (task) frame of the robot, and $K^{\mathrm{ca}}\in\mathbb{R}^{m\times
m}$ and $D^{\mathrm{ca}}\in\mathbb{R}^{m\times m}$ being the virtual Cartesian
stiffness and damping matrices, respectively.

$\tau_{\mathrm{c}}^{\mathrm{ns}}$ is the torque commanded to achieve a joint
impedance behavior with respect to a desired configuration and projected in
the null-space of the robot’s Jacobian, to not affect the Cartesian motion of
the robot’s end effector.

$\tau_{\mathrm{c}}^{\mathrm{ext}}$ is the torque commanded to achieve the
desired external force command in the frame of the end effector of the robot,
$F_{\mathrm{c}}^{\mathrm{ext}}$:

$\tau_{\mathrm{c}}^{\mathrm{ext}}=J^{\mathrm{T}}(q)F_{\mathrm{c}}^{\mathrm{ext}}$
(4)

In this work we utilize $\tau_{\mathrm{c}}^{\mathrm{ext}}$ and
$\tau_{\mathrm{c}}^{\mathrm{ca}}$ in combination with changing stiffnesses by
passing configurations through skill primitives.

#### Cartesian Compliance Control

In contrast to the KUKA iiwa, the Universal Robots UR5e like many other
industrial manipulators does not support direct torque control. This means
that a compliant control solution is needed that can work with joint position
or joint velocity commands. Recently, such a method was introduced through
forward dynamics compliance control [38]. It combines admittance, impedance
and force control into one control strategy. The control loop is closed only
through a force-torque sensor, which makes it system independent. Internally
it utilizes a forward dynamics simulation of a virtual model of the robot
system to map Cartesian inputs to joint commands. For further details, we
refer the interested reader to [38] and [39]. Like the impedance controller
that was introduced in the previous section, this controller solution also
allows for the runtime modification of applied external forces, reference
poses and stiffness changes. We utilize these capabilities when modeling
behaviors with BTs.

##  V Case Study: Wiping Task

We demonstrate the usage of skills in different tasks and on different robot
system with the example of a wiping task.

Figure 4: The iiwa wiping task on a table. The different hardware needs a
different controller setup and gripper skills.

Figure 5: The resulting reference positions and actual positions of the end
effectors in a) the whiteboard wiping task and with the _UR5e_ b) the table
wiping task with the _iiwa_. The shaded green shows the surface area that has
been covered by the eraser that is used for cleaning.

###  V-A Challenges

The task at hand and the robot systems selected for this study present a range
of intriguing challenges. Firstly, the task instances involve surfaces with
different properties, namely a smooth whiteboard and a rough industrial table.
The whiteboard is mounted to a wall, whereas the table surface is parallel to
the floor. Moreover, we assume that the precise distance between the robot and
the surface may not be known in advance. The robot systems are equipped with
different grippers, each of which has its own communication interface.
Finally, the robots have distinct kinematics, vendor-supplied programming
solutions, and control interfaces. In this context, we demonstrate how a skill
that utilizes knowledge representation, planning and the automatic skill
implementation selection can be executed in different context on different
robot systems.

###  V-B Implementation

The implementation of such a skill incorporates several key aspects that are
discussed in [23]. Specifically, we utilize a motion generator (MG) that
comprises of both trajectory generation (Sec.IV-B) and a controller
(Sec.IV-C).

As inputs for the skill, only the robot arm and a surface to clean need to be
selected among the available instances in the WM. All other necessary data,
such as robot-specific properties (e.g., the controller to use) and the
surface properties (e.g. the dimensions, force to apply) are automatically
fetched from the WM as part of the knowledge integration. Apart from the
parameters, this skill also contains the necessary pre- and postconditions for
planning. The integrated plannerin SkiROS can receive a goal such as
(skiros:clean scalable:Workstation-1186 scalable:Cell-12). For instance, one
precondition of the skill is that the tool must already be held in the
gripper. This is leveraged in task-level planning, where the robot is required
to pick up the tool first. In case of a mobile system, this may entail driving
to a different location. The postcondition of the task is that the surface is
cleaned.

To begin the wiping process, the end effector is moved to the corner of the
surface to be cleaned. Once there, it disables the stiffness of the compliant
controller along the normal of the surface and applies a predetermined force.
In our experiments, $\mathrm{8}$ are specified for surfaces that are easier to
clean, such as the whiteboard, and $\mathrm{1}\mathrm{8}$ for the table. The
robot then applies an overlay motion, in this case a circular motion, while
performing the wiping action. From there the skill executes laps by moving the
robot’s end-effector to the right, then down, and finally left until the
entire surface is covered. Finally, upon completion of the task, the robot
stops the force application, increases the stiffness along the surface normal,
and moves the end-effector away from the surface.

###  V-C Experiments and Discussion

We conducted the experiments using two different robot systems. The first
system is a mobile platform that has a UR5e 6 degree-of-freedom robot arm with
a Schunk two finger gripper. The second system is a 7 degree-of-freedom KUKA
iiwa with a _Robotiq_ 3-finger gripper. We selected two different surfaces for
the cleaning task, each with different orientations and surface properties.

The resulting reference path and the actual path for the _UR5e_ are shown in
Fig. 5a). The implementation spends more time in the left and right extremes
since the sub-trajectory ends there and it switches to new lane. This can be
adressed by blending between trajectories. During the execution we also
noticed subtle vibrations, especially when the arm was stretched out.

In our second experiment we used the KUKA iiwa robot system to wipe the
surface of a table. As in the previous experiment, the important attributes
were represented in the WM, and the skill could be started by simply selecting
the surface and the robot arm. The reference path and the actual path for the
robot end effector during the task is shown in Figure 5b). The controller is
very stable and there are no vibrations. We observed that the eraser was
occasionally lifted slightly during the wiping process. This could potentially
be addressed by increasing the rotational stiffness or applying more pressure
to the surface.

In both tasks, the respective robot systems were able to completely clean the
surface with the proposed implementation. In principle, the surface can also
be cleaned using only the Cartesian linear motion. However, as intuitively
expected, the addition of an additional overlay motion improves the cleaning
performance in practice.

##  VI Conclusions and Future Work

The transition towards Industry 4.0 brings challenges such as small batch
sizes, constantly changing tasks and environments. Together with the
transition towards human-robot colaboration and handling contact-rich tasks,
it creates the need for platforms that can address these challenges.

In this paper we presented a complete architecture that allows to transfer
skills between different robot systems. We explain the necessary prerequisites
and knowledge representation. We demonstrated this by successfully performing
contact-rich wiping tasks with a _KUKA iiwa_ and a _Universal Robots UR5e_.
All essential building blocks including the skill-based system and the
controllers are available as open-source software and the results are expected
to scale to other hardware and tasks as well.

While the wiping performance was sufficient in our tasks, it can potentially
be increased by learning a good combination of path velocities and applied
force. A visual inspection with a camera is a next inuitive step. We are also
looking into an evaluation with other platforms such as the Franka Emika Robot
(Panda).

## ACKNOWLEDGMENT

This work was partially supported by the Wallenberg AI, Autonomous Systems and
Software Program (WASP) funded by Knut and Alice Wallenberg Foundation.

## References

  * [1] R. Volpe, I. Nesnas, T. Estlin, D. Mutz, R. Petras, and H. Das, “The claraty architecture for robotic autonomy,” in _2001 IEEE Aerospace Conference Proceedings (Cat. No. 01TH8542)_ , vol. 1. IEEE, 2001, pp. 1–121.
  * [2] S. Bensalem, M. Gallien, F. Ingrand, I. Kahloul, and N. Thanh-Hung, “Designing autonomous robots,” _IEEE Robotics & Automation Magazine_, vol. 16, no. 1, pp. 67–77, 2009.
  * [3] M. Stenmark and J. Malec, “Knowledge-Based Industrial Robotics,” _Frontiers in Artificial Intelligence and Applications_ , pp. 265–274, 2013\.
  * [4] ——, “Knowledge-based instruction of manipulation tasks for industrial robotics,” _Robotics and Computer-Integrated Manufacturing_ , vol. 33, pp. 56–67, Jun. 2015.
  * [5] M. Tenorth and M. Beetz, “KNOWROB — knowledge processing for autonomous personal robots,” in _2009 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , Oct. 2009, pp. 4261–4266.
  * [6] ——, “KnowRob: A knowledge processing infrastructure for cognition-enabled robots,” _The International Journal of Robotics Research_ , vol. 32, no. 5, pp. 566–590, Apr. 2013.
  * [7] E. A. Topp, M. Stenmark, A. Ganslandt, A. Svensson, M. Haage, and J. Malec, “Ontology-based knowledge representation for increased skill reusability in industrial robots,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2018, pp. 5672–5678.
  * [8] D. Wuthier, F. Rovida, M. Fumagalli, and V. Krüger, “Productive multitasking for industrial robots,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 12 654–12 661.
  * [9] J. L. Wyatt, A. Aydemir, M. Brenner, M. Hanheide, N. Hawes, P. Jensfelt, M. Kristan, G.-J. M. Kruijff, P. Lison, A. Pronobis _et al._ , “Self-understanding and self-extension: A systems and representational approach,” _IEEE Transactions on Autonomous Mental Development_ , vol. 2, no. 4, pp. 282–303, 2010.
  * [10] D. Stampfer, A. Lotz, M. Lutz, and C. Schlegel, “The smartmdsd toolchain: An integrated mdsd workflow and integrated development environment (ide) for robotics software,” _Journal of Software Engineering for Robotics (JOSER)_ , vol. 7, no. 1, pp. 3–19, 2016.
  * [11] C. Paxton, A. Hundt, F. Jonathan, K. Guerin, and G. D. Hager, “Costar: Instructing collaborative robots with behavior trees and vision,” in _2017 IEEE international conference on robotics and automation (ICRA)_. IEEE, 2017, pp. 564–571.
  * [12] B. Bruno, N. Y. Chong, H. Kamide, S. Kanoria, J. Lee, Y. Lim, A. K. Pandey, C. Papadopoulos, I. Papadopoulos, F. Pecora _et al._ , “The caresses eu-japan project: making assistive robots culturally competent,” in _Ambient Assisted Living: Italian Forum 2017 8_. Springer, 2019, pp. 151–169.
  * [13] B. Gibaud, G. Forestier, C. Feldmann, G. Ferrigno, P. Gonçalves, T. Haidegger, C. Julliard, D. Katić, H. Kenngott, L. Maier-Hein _et al._ , “Toward a standard ontology of surgical process models,” _International journal of computer assisted radiology and surgery_ , vol. 13, pp. 1397–1408, 2018.
  * [14] H. Azevedo, J. P. R. Belo, and R. A. Romero, “Ontpercept: a perception ontology for robotic systems,” in _2018 Latin American Robotic Symposium, 2018 Brazilian Symposium on Robotics (SBR) and 2018 Workshop on Robotics in Education (WRE)_. IEEE, 2018, pp. 469–475.
  * [15] S. Manzoor, Y. G. Rocha, S.-H. Joo, S.-H. Bae, E.-J. Kim, K.-J. Joo, and T.-Y. Kuc, “Ontology-based knowledge representation in robotic systems: a survey oriented toward applications,” _Applied Sciences_ , vol. 11, no. 10, p. 4324, 2021.
  * [16] A. Ijspeert, J. Nakanishi, and S. Schaal, “Trajectory formation for imitation with nonlinear dynamical systems,” in _Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180)_ , vol. 2, 2001, pp. 752–757 vol.2.
  * [17] A. J. Ijspeert, J. Nakanishi, and S. Schaal, “Learning rhythmic movements by demonstration using nonlinear oscillators,” in _Proceedings of the ieee/rsj int. conference on intelligent robots and systems (iros2002)_ , no. CONF, 2002, pp. 958–963.
  * [18] A. Paraschos, C. Daniel, J. R. Peters, and G. Neumann, “Probabilistic movement primitives,” _Advances in neural information processing systems_ , vol. 26, 2013.
  * [19] D. Leidner, C. Borst, A. Dietrich, M. Beetz, and A. Albu-Schäffer, “Classifying compliant manipulation tasks for automated planning in robotics,” in _2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2015, pp. 1769–1776.
  * [20] D. Leidner, W. Bejjani, A. Albu-Schäffer, and M. Beetz, “Robotic agents representing, reasoning, and executing wiping tasks for daily household chores,” _AUTONOMOUS AGENTS AND MULTI-AGENT SYSTEMS_ , 2016.
  * [21] D. Leidner, G. Bartels, W. Bejjani, A. Albu-Schäffer, and M. Beetz, “Cognition-enabled robotic wiping: Representation, planning, execution, and interpretation,” _Robotics and Autonomous Systems_ , vol. 114, pp. 199–216, 2019.
  * [22] F. Rovida, B. Grossmann, and V. Krüger, “Extended behavior trees for quick definition of flexible robotic tasks,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2017, pp. 6793–6800.
  * [23] F. Rovida, D. Wuthier, B. Grossmann, M. Fumagalli, and V. Krüger, “Motion generators combined with behavior trees: A novel approach to skill modelling,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2018, pp. 5964–5971.
  * [24] M. Mayr, F. Ahmad, K. Chatzilygeroudis, L. Nardi, and V. Krueger, “Combining planning, reasoning and reinforcement learning to solve industrial robot tasks,” _IROS 2022 Workshop on Trends and Advances in Machine Learning and Automated Reasoning for Intelligent Robots and Systems_ , 2022.
  * [25] ——, “Skill-based multi-objective reinforcement learning of industrial robot tasks with planning and knowledge integration,” in _2022 IEEE International Conference on Robotics and Biomimetics (ROBIO)_. IEEE, 2022, pp. 1995–2002.
  * [26] M. Mayr, C. Hvarfner, K. Chatzilygeroudis, L. Nardi, and V. Krueger, “Learning skill-based industrial robot tasks with user priors,” in _2022 IEEE 18th International Conference on Automation Science and Engineering (CASE)_. IEEE, 2022, pp. 1485–1492.
  * [27] F. Ahmad, M. Mayr, and V. Krueger, “Learning to adapt the parameters of behavior trees and motion generators to task variations,” 2023.
  * [28] C. Aeronautiques, A. Howe, C. Knoblock, I. D. McDermott, A. Ram, M. Veloso, D. Weld, D. W. SRI, A. Barrett, D. Christianson _et al._ , “Pddl| the planning domain definition language,” _Technical Report, Tech. Rep._ , 1998\.
  * [29] P. Eyerich, R. Mattmüller, and G. Röger, “Using the context-enhanced additive heuristic for temporal and numeric planning,” in _Nineteenth International Conference on Automated Planning and Scheduling_ , 2009.
  * [30] S. Bøgh, O. S. Nielsen, M. R. Pedersen, V. Krüger, and O. Madsen, “Does your robot have skills?” in _Proceedings of the 43rd international symposium on robotics_. VDE Verlag GMBH, 2012.
  * [31] M. Colledanchise and P. Ögren, _Behavior Trees in Robotics and AI: An Introduction_. Chapman & Hall/CRC Press, 2017.
  * [32] ——, “How Behavior Trees modularize robustness and safety in hybrid systems,” in _2014 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , 2014, pp. 1482–1488.
  * [33] M. Iovino, E. Scukins, J. Styrud, P. Ögren, and C. Smith, “A survey of behavior trees in robotics and ai,” 2020.
  * [34] A. Marzinotto, M. Colledanchise, C. Smith, and P. Ögren, “Towards a unified behavior trees framework for robot control,” in _2014 IEEE International Conference on Robotics and Automation (ICRA)_ , 2014, pp. 5420–5427.
  * [35] F. Ahmad, M. Mayr, E. A. Topp, J. Malec, and V. Krueger, “Generalizing behavior trees and motion-generator (btmg) policy representation for robotic tasks over scenario parameters,” in _2022 IJCAI Planning and Reinforcement Learning Workshop_ , 2022.
  * [36] M. Mayr, K. Chatzilygeroudis, F. Ahmad, L. Nardi, and V. Krueger, “Learning of Parameters in Behavior Trees for Movement Skills,” in _2021 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , 2021.
  * [37] M. Mayr and J. M. Salt-Ducaju, “A c++ implementation of a cartesian impedance controller for robotic manipulators,” _arXiv preprint arXiv:2212.11215_ , 2022.
  * [38] S. Scherzinger, A. Roennau, and R. Dillmann, “Forward dynamics compliance control (fdcc): A new approach to cartesian compliance for robotic manipulators,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2017, pp. 4568–4575.
  * [39] ——, “Inverse kinematics with forward dynamics solvers for sampled motion tracking,” in _2019 19th International Conference on Advanced Robotics (ICAR)_. IEEE, 2019, pp. 681–687.

  *[GUI]: graphical user interface
  *[RL]: reinforcement learning
  *[WM]: world model
  *[RDF]: Resource Description Framework
  *[PDDL]: Planning Domain Definition Language
  *[MES]: manufacturing execution system
  *[BT]: behavior tree
  *[OWL]: Web Ontology Language
  *[ROS]: Robot Operating System
  *[API]: application programming interface
  *[MG]: motion generator


PROMISE: Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates

Zachary Frangella^*<EMAIL_ADDRESS>
Stanford University

Pratik Rathore^*<EMAIL_ADDRESS>
Stanford University

Shipu Zhao<EMAIL_ADDRESS>
Cornell University
Madeleine Udell<EMAIL_ADDRESS>
Stanford University

This paper introduces PROMISE (Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates), a suite of sketching-based preconditioned stochastic gradient algorithms for solving large-scale convex optimization problems arising in machine learning.
PROMISE includes preconditioned versions of SVRG, SAGA, and Katyusha;
each algorithm comes with a strong theoretical analysis and
effective default hyperparameter values.
In contrast, traditional stochastic gradient methods
require careful hyperparameter tuning to succeed,
and degrade in the presence of ill-conditioning,
a ubiquitous phenomenon in machine learning.
Empirically, we verify the superiority of the proposed algorithms
by showing that, using default hyperparameter values,
they outperform or match popular tuned stochastic gradient optimizers
on a test bed of $51$ ridge and logistic regression problems
assembled from benchmark machine learning repositories.
On the theoretical side, this paper introduces the notion of quadratic regularity
in order to establish linear convergence of all proposed methods
even when the preconditioner is updated infrequently.
The speed of linear convergence is determined by the quadratic regularity ratio,
which often provides a tighter bound on the convergence rate compared to the condition number,
both in theory and in practice,
and explains the fast global linear convergence of the proposed methods.

Stochastic optimization, quasi-Newton, preconditioning, randomized low-rank approximation

§ TABLE OF CONTENTS

[Alaoui and Mahoney, 2015]
Ahmed Alaoui and Michael W Mahoney.
Fast Randomized Kernel Ridge Regression with Statistical
In Advances in Neural Information Processing Systems, 2015.

[Allen-Zhu, 2018]
Zeyuan Allen-Zhu.
Katyusha: The First Direct Acceleration of Stochastic
Gradient Methods.
Journal of Machine Learning Research, 180
(221):0 1–51, 2018.

[Arjevani and Shamir, 2017]
Yossi Arjevani and Ohad Shamir.
Oracle complexity of second-order methods for finite-sum problems.
In International Conference on Machine Learning, pages
205–213. PMLR, 2017.

[Bach, 2010]
Francis Bach.
Self-concordant analysis for logistic regression.
Electronic Journal of Statistics, 2010.

[Bach, 2013]
Francis Bach.
Sharp analysis of low-rank kernel matrix approximations.
In Conference on Learning Theory, 2013.

[Beck, 2017]
Amir Beck.
First-Order Methods in Optimization.
Society for Industrial and Applied Mathematics, Philadelphia, PA,

[Bollapragada et al., 2018]
Raghu Bollapragada, Jorge Nocedal, Dheevatsa Mudigere, Hao-Jun Shi, and Ping
Tak Peter Tang.
A progressive batching L-BFGS method for machine learning.
In International Conference on Machine Learning, pages
620–629. PMLR, 2018.

[Bollapragada et al., 2019]
Raghu Bollapragada, Richard H Byrd, and Jorge Nocedal.
Exact and inexact subsampled Newton methods for optimization.
IMA Journal of Numerical Analysis, 390 (2):0
545–578, 2019.

[Boyd and Vandenberghe, 2004]
Stephen P Boyd and Lieven Vandenberghe.
Convex optimization.
Cambridge University Press, 2004.

[Byrd et al., 2011]
Richard H Byrd, Gillian M Chin, Will Neveitt, and Jorge Nocedal.
On the use of stochastic Hessian information in optimization
methods for machine learning.
SIAM Journal on Optimization, 210 (3):0
977–995, 2011.

[Byrd et al., 2016]
Richard H Byrd, Samantha L Hansen, Jorge Nocedal, and Yoram Singer.
A stochastic quasi-Newton method for large-scale optimization.
SIAM Journal on Optimization, 260 (2):0
1008–1031, 2016.

[Candes and Recht, 2012]
Emmanuel Candes and Benjamin Recht.
Exact matrix completion via convex optimization.
Communications of the ACM, 550 (6):0 111–119,

[Candes and Romberg, 2007]
Emmanuel Candes and Justin Romberg.
Sparsity and incoherence in compressive sampling.
Inverse problems, 230 (3):0 969, 2007.

[Caponnetto and De Vito, 2007]
Andrea Caponnetto and Ernesto De Vito.
Optimal rates for the regularized least-squares algorithm.
Foundations of Computational Mathematics, 7:0
331–368, 2007.

[Chang and Lin, 2011]
Chih-Chung Chang and Chih-Jen Lin.
LIBSVM: A library for support vector machines.
ACM Transactions on Intelligent Systems and Technology,
2:0 27:1–27:27, 2011.
Software available at <http://www.csie.ntu.edu.tw/ cjlin/libsvm>.

[Cohen et al., 2016]
Michael B Cohen, Jelani Nelson, and David P Woodruff.
Optimal Approximate Matrix Product in Terms of Stable
In 43rd International Colloquium on Automata, Languages, and
Programming, 2016.

[Cohen et al., 2017]
Michael B Cohen, Cameron Musco, and Christopher Musco.
Input sparsity time low-rank approximation via ridge leverage score
In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium
on Discrete Algorithms, pages 1758–1777. SIAM, 2017.

[Defazio et al., 2014]
Aaron Defazio, Francis Bach, and Simon Lacoste-Julien.
SAGA: A fast incremental gradient method with support for
non-strongly convex composite objectives.
Advances in Neural Information Processing Systems, 27, 2014.

[Dereziński, 2022]
Michał Dereziński.
Stochastic Variance-Reduced Newton: Accelerating
Finite-Sum Minimization with Large Batches.
arXiv preprint arXiv:2206.02702, 2022.

[Derezinski et al., 2020]
Michal Derezinski, Feynman T Liang, Zhenyu Liao, and Michael W Mahoney.
Precise expressions for random projections: Low-rank approximation
and randomized Newton.
In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin,
editors, Advances in Neural Information Processing Systems, volume 33,
pages 18272–18283. Curran Associates, Inc., 2020.

[Derezinski et al., 2021]
Michal Derezinski, Jonathan Lacotte, Mert Pilanci, and Michael W Mahoney.
Newton-LESS: Sparsification without Trade-offs for the
Sketched Newton Update.
In Advances in Neural Information Processing Systems,
volume 34, 2021.

[Díaz et al., 2023]
Mateo Díaz, Ethan N Epperly, Zachary Frangella, Joel A Tropp, and Robert J
Robust, randomized preconditioning for kernel ridge regression.
arXiv preprint arXiv:2304.12465, 2023.

[Doikov et al., 2023]
Nikita Doikov, El Mahdi Chayti, and Martin Jaggi.
Second-order optimization with lazy Hessians.
In International Conference on Machine Learning, pages
8138–8161. PMLR, 2023.

[Erdogdu and Montanari, 2015]
Murat A Erdogdu and Andrea Montanari.
Convergence rates of sub-sampled Newton methods.
Advances in Neural Information Processing Systems, 28, 2015.

[Frangella et al., 2023]
Zachary Frangella, Pratik Rathore, Shipu Zhao, and Madeleine Udell.
SketchySGD: Reliable Stochastic Optimization via Randomized
Curvature Estimates, 2023a.

[Frangella et al., 2023]
Zachary Frangella, Joel A. Tropp, and Madeleine Udell.
Randomized Nyström Preconditioning.
SIAM Journal on Matrix Analysis and Applications, 440
(2):0 718–752, 2023b.

[Gazagnadou et al., 2019]
Nidham Gazagnadou, Robert Gower, and Joseph Salmon.
Optimal Mini-Batch and Step Sizes for SAGA.
In Proceedings of the 36th International Conference on Machine
Learning, Proceedings of Machine Learning Research, pages 2142–2150. PMLR,
09–15 Jun 2019.

[Gittens and Mahoney, 2016]
Alex Gittens and Michael W Mahoney.
Revisiting the Nyström method for improved large-scale machine
The Journal of Machine Learning Research, 170
(1):0 3977–4041, 2016.

[Gonen et al., 2016]
Alon Gonen, Francesco Orabona, and Shai Shalev-Shwartz.
Solving ridge regression using sketched preconditioned SVRG.
In International Conference on Machine Learning, pages
1397–1405. PMLR, 2016.

[Gower et al., 2016]
Robert Gower, Donald Goldfarb, and Peter Richtárik.
Stochastic block BFGS: Squeezing more curvature out of data.
In International Conference on Machine Learning, pages
1869–1878. PMLR, 2016.

[Gower et al., 2018]
Robert Gower, Nicolas Le Roux, and Francis Bach.
Tracking the gradients using the Hessian: A new look at variance
reducing stochastic methods.
In International Conference on Artificial Intelligence and
Statistics, pages 707–715. PMLR, 2018.

[Gower et al., 2019]
Robert Gower, Dmitry Kovalev, Felix Lieder, and Peter Richtárik.
RSN: Randomized subspace Newton.
Advances in Neural Information Processing Systems, 32,

[Gower et al., 2020]
Robert M Gower, Mark Schmidt, Francis Bach, and Peter Richtárik.
Variance-reduced methods for machine learning.
Proceedings of the IEEE, 1080 (11):0
1968–1983, 2020.

[Gower et al., 2019]
Robert Mansel Gower, Nicolas Loizou, Xun Qian, Alibek Sailanbayev, Egor
Shulgin, and Peter Richtárik.
SGD: General analysis and improved rates.
In International Conference on Machine Learning, pages
5200–5209. PMLR, 2019b.

[Higham, 2002]
Nicholas J Higham.
Accuracy and stability of numerical algorithms.
SIAM, 2002.

[Hsu et al., 2014]
Daniel Hsu, Sham M Kakade, and Tong Zhang.
Random Design Analysis of Ridge Regression.
Foundations of Computational Mathematics, 30
(14):0 569–600, 2014.

[Jambulapati et al., 2020]
Arun Jambulapati, Jerry Li, Christopher Musco, Aaron Sidford, and Kevin Tian.
Fast and near-optimal diagonal preconditioning.
arXiv preprint arXiv:2008.01722, 2020.

[Johnson and Zhang, 2013]
Rie Johnson and Tong Zhang.
Accelerating stochastic gradient descent using predictive variance
Advances in Neural Information Processing Systems, 26, 2013.

[Karimireddy et al., 2018]
Sai Praneeth Karimireddy, Sebastian U Stich, and Martin Jaggi.
Global linear convergence of Newton's method without
strong-convexity or Lipschitz gradients.
arXiv preprint arXiv:1806.00413, 2018.

[Kovalev et al., 2019]
Dmitry Kovalev, Konstantin Mishchenko, and Peter Richtárik.
Stochastic Newton and cubic Newton methods with simple local
linear-quadratic rates.
arXiv preprint arXiv:1912.01597, 2019.

[Kovalev et al., 2020]
Dmitry Kovalev, Samuel Horváth, and Peter Richtárik.
Don’t Jump Through Hoops and Remove Those Loops: SVRG
and Katyusha are Better Without the Outer Loop.
In Aryeh Kontorovich and Gergely Neu, editors, Proceedings of
the 31st International Conference on Algorithmic Learning Theory, volume 117
of Proceedings of Machine Learning Research, pages 451–467. PMLR, 08
Feb–11 Feb 2020.

[Lacotte et al., 2021]
Jonathan Lacotte, Yifei Wang, and Mert Pilanci.
Adaptive Newton sketch: linear-time optimization with quadratic
convergence and effective Hessian dimensionality.
In International Conference on Machine Learning, pages
5926–5936. PMLR, 2021.

[Li et al., 2020]
Xiang Li, Shusen Wang, and Zhihua Zhang.
Do Subsampled Newton Methods work for High-Dimensional
In Proceedings of the AAAI Conference on Artificial
Intelligence, volume 34, pages 4723–4730, 2020.

[Liu et al., 2019]
Yanli Liu, Fei Feng, and Wotao Yin.
Acceleration of SVRG and Katyusha X by inexact preconditioning.
In International Conference on Machine Learning, pages
4003–4012. PMLR, 2019.

[Marteau-Ferey et al., 2019]
Ulysse Marteau-Ferey, Francis Bach, and Alessandro Rudi.
Globally convergent Newton methods for ill-conditioned generalized
self-concordant losses.
Advances in Neural Information Processing Systems, 32,

[Marteau-Ferey et al., 2019]
Ulysse Marteau-Ferey, Dmitrii Ostrovskii, Francis Bach, and Alessandro Rudi.
Beyond least-squares: Fast rates for regularized empirical risk
minimization through self-concordance.
In Conference on Learning Theory, pages 2294–2340. PMLR,

[Martinsson and Tropp, 2020]
Per-Gunnar Martinsson and Joel A Tropp.
Randomized numerical linear algebra: Foundations and algorithms.
Acta Numerica, 29:0 403–572, 2020.

[Mei and Montanari, 2022]
Song Mei and Andrea Montanari.
The Generalization Error of Random Features Regression:
Precise Asymptotics and the Double Descent Curve.
Communications on Pure and Applied Mathematics, 750
(4):0 667–766, 2022.

[Milman and Schechtman, 2009]
Vitali D Milman and Gideon Schechtman.
Asymptotic theory of finite dimensional normed spaces:
Isoperimetric inequalities in Riemannian manifolds, volume 1200.
Springer, 2009.

[Minsker, 2017]
Stanislav Minsker.
On some extensions of Bernstein’s inequality for self-adjoint
Statistics & Probability Letters, 127:0 111–119,

[Moritz et al., 2016]
Philipp Moritz, Robert Nishihara, and Michael Jordan.
A linearly-convergent stochastic L-BFGS algorithm.
In Artificial Intelligence and Statistics, pages 249–258.
PMLR, 2016.

[Moulines and Bach, 2011]
Eric Moulines and Francis Bach.
Non-asymptotic analysis of stochastic approximation algorithms for
machine learning.
Advances in Neural Information Processing Systems, 24, 2011.

[Musco and Musco, 2015]
Cameron Musco and Christopher Musco.
Randomized block Krylov methods for stronger and faster approximate
singular value decomposition.
Advances in Neural Information Processing Systems, 28, 2015.

[Na et al., 2022]
Sen Na, Michał Dereziński, and Michael W Mahoney.
Hessian averaging in stochastic Newton methods achieves superlinear
Mathematical Programming, pages 1–48, 2022.

[Nemirovski et al., 2009]
Arkadi Nemirovski, Anatoli Juditsky, Guanghui Lan, and Alexander Shapiro.
Robust stochastic approximation approach to stochastic programming.
SIAM Journal on optimization, 190 (4):0
1574–1609, 2009.

[Nemirovski and Yudin, 1983]
Arkadi S Nemirovski and David B Yudin.
Problem complexity and method efficiency in optimization.
Wiley-Interscience, 1983.

[Nesterov, 2018]
Yurii Nesterov.
Lectures on convex optimization, volume 137.
Springer, 2018.

[Nocedal and Wright, 1999]
Jorge Nocedal and Stephen J Wright.
Numerical optimization.
Springer, 1999.

[Pedregosa et al., 2011]
Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel,
Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron
Weiss, Vincent Dubourg, et al.
Scikit-learn: Machine learning in Python.
the Journal of machine Learning Research, 12:0
2825–2830, 2011.

[Pilanci and Wainwright, 2017]
Mert Pilanci and Martin J Wainwright.
Newton Sketch: A near linear-time optimization algorithm with
linear-quadratic convergence.
SIAM Journal on Optimization, 270 (1):0
205–245, 2017.

[Qu et al., 2022]
Zhaonan Qu, Wenzhi Gao, Oliver Hinder, Yinyu Ye, and Zhengyuan Zhou.
Optimal diagonal preconditioning: Theory and practice.
arXiv preprint arXiv:2209.00809, 2022.

[Rahimi and Recht, 2007]
Ali Rahimi and Benjamin Recht.
Random features for large-scale kernel machines.
Advances in Neural Information Processing Systems, 20, 2007.

[Roosta-Khorasani and Mahoney, 2019]
Farbod Roosta-Khorasani and Michael W Mahoney.
Sub-sampled Newton methods.
Mathematical Programming, 1740 (1):0 293–326,

[Rudelson and Vershynin, 2013]
Mark Rudelson and Roman Vershynin.
Hanson-Wright inequality and sub-gaussian concentration.
Electronic Communications in Probability, 2013.

[Rudi et al., 2017]
Alessandro Rudi, Luigi Carratino, and Lorenzo Rosasco.
Falkon: An optimal large scale kernel method.
Advances in Neural Information Processing Systems, 30, 2017.

[Rudi et al., 2018]
Alessandro Rudi, Daniele Calandriello, Luigi Carratino, and Lorenzo Rosasco.
On fast leverage score sampling and optimal learning.
Advances in Neural Information Processing Systems, 31, 2018.

[Sutherland, 2017]
Danica J Sutherland.
Fixing an error in Caponnetto and de Vito (2007).
arXiv preprint arXiv:1702.02982, 2017.

[Tropp et al., 2017]
Joel A Tropp, Alp Yurtsever, Madeleine Udell, and Volkan Cevher.
Fixed-rank approximation of a positive-semidefinite matrix from
streaming data.
Advances in Neural Information Processing Systems, 30, 2017.

[Tropp et al., 2019]
Joel A. Tropp, Alp Yurtsever, Madeleine Udell, and Volkan Cevher.
Streaming Low-Rank Matrix Approximation with an Application
to Scientific Simulation.
SIAM Journal on Scientific Computing, 410
(4):0 A2430–A2463, 2019.

[Vanschoren et al., 2013]
Joaquin Vanschoren, Jan N. van Rijn, Bernd Bischl, and Luis Torgo.
OpenML: Networked Science in Machine Learning.
SIGKDD Explorations, 150 (2):0 49–60, 2013.

[Williams and Seeger, 2000]
Christopher Williams and Matthias Seeger.
Using the Nyström method to speed up kernel machines.
Advances in Neural Information Processing Systems, 13, 2000.

[Woodworth and Srebro, 2016]
Blake E Woodworth and Nati Srebro.
Tight complexity bounds for optimizing composite objectives.
Advances in Neural Information Processing Systems, 29, 2016.

[Ye et al., 2021]
Haishan Ye, Luo Luo, and Zhihua Zhang.
Approximate Newton Methods.
Journal of Machine Learning Research, 220
(66):0 1–41, 2021.
URL <http://jmlr.org/papers/v22/19-870.html>.

[Yelp, 2023]
Yelp dataset, 2023.
URL <https://www.yelp.com/dataset>.

[Zhao et al., 2022]
Shipu Zhao, Zachary Frangella, and Madeleine Udell.
NysADMM: faster composite convex optimization via low-rank
In Proceedings of the 39th International Conference on Machine
Learning, volume 162 of Proceedings of Machine Learning Research,
pages 26824–26840. PMLR, 17–23 Jul 2022.


intelligent systems that assist in skill development and strategic improvement
in chess.

  7. 7.

Adversarial Attacks in Chess: With the increasing use of language models in
critical applications like chess analysis and decision-making, it becomes
essential to investigate potential vulnerabilities and develop defenses
against adversarial attacks. Researchers can explore techniques to generate
adversarial examples specifically targeted at chess-related tasks. By
identifying weaknesses in language models’ state tracking or policy evaluation
abilities, researchers can enhance the robustness and security of these
models.

  8. 8.

Chess Game Generation: Researchers can utilize the provided dataset to develop
models capable of generating new chess game sequences. By leveraging the
language model’s understanding of chess moves and game structures, researchers
can explore generative models that can produce novel and diverse chess game
sequences. This can be beneficial for various applications, including chess
game analysis, training data generation, and even game generation for chess
variants.

Overall, our dataset and benchmark offer numerous potential directions,
ranging from dataset expansion and transfer learning to exploring chess
variants, education, analysis, and game generation. These directions have the
potential to advance the field of chess-related language modeling and provide
valuable tools and resources for chess players and enthusiasts.

##  Appendix K Limitations and Potential Societal Impact

The availability of a comprehensive and diverse chess dataset presents both
limitations and potential societal impacts that researchers should consider.

Limitations. While the chess dataset provided in this study is valuable, it is
important to acknowledge its limitations. One limitation is the potential bias
introduced by relying on historical Lichess matches from different time
periods. This may result in variations in player strategies, popular openings,
and game trends over time, potentially impacting the generalizability of the
dataset. Additionally, it is worth noting that the dataset predominantly
focuses on standard chess, and may not encompass the full spectrum of chess
variants. Researchers interested in exploring niche or less-popular variants
may need to gather additional data from specific sources to ensure
comprehensive coverage of different chess variants. These considerations are
crucial to ensure the validity and applicability of research findings based on
the provided dataset.

Potential Societal Impact. The availability of a comprehensive and diverse
chess dataset can have a significant societal impact. First and foremost, it
can contribute to the development of more advanced and intelligent chess-
playing agents. These agents can be utilized in various applications, such as
chess analysis, training tools for players of different skill levels, and even
as opponents for chess enthusiasts. The dataset can also facilitate the
advancement of chess education by providing valuable resources for tutorials,
interactive learning platforms, and strategic guidance. Additionally, the
dataset can inspire research in the field of artificial intelligence,
contributing to the development of innovative techniques that can be applied
beyond the domain of chess. Lastly, the dataset can encourage the exploration
of explainable AI methods in chess, enabling players to understand and learn
from the reasoning behind the model’s moves, thereby promoting transparency
and trust in AI systems.