* $\widehat{\Gbf}^{(m)}_{ksj} = \lan \partial^{\alpha^s}\psi_k, f_j(\tilde{\Ubf}^{(m),\star}+\widehat{\ep}^{(m)})\ran_m$

* $\overline{\Gbf}^{(m)}_{ksj} = \lan \partial^{\alpha^s}\psi_k, \rho^{(m)}\star f_j(\tilde{\Ubf}^{(m),\star})\ran_m$

* $\overline{\Gbf}^{(m),\star}_{ksj} = \lan \partial^{\alpha^s}\psi_k, f_j(\tilde{\Ubf}^{(m),\star})\ran_m$

* $\Gbf^{(m),\star}_{ksj} = \lan \partial^{\alpha^s}\psi_k, f_j(\Ubf^{(m),\star})\ran_m$

* $\Gbf^\star_{ksj} = \lan \partial^{\alpha^s}\psi_k, f_j(u)\ran$.

$I_4$: First, we note that $I_4=\|\overline{\Gbf}^{(m),\star} - \Gbf^{(m),\star}\|$ is determined by how fast the locally averaged clean data $\tilde{\Ubf}^{(m),\star}$ converges to the clean data $\Ubf^{(m),\star}$. The locally Lipschitz assumption on the library and the smoothness of $u\in \CalH^{k,\infty}$ with $k>\frac{d+1}{2}$ implies that there exists $\tilde{C}$ depending on $C_\CalF$, $\psi$, and $\nrm{u}_\infty$ such that $m$ large enough implies

\[I_4 \leq \tilde{C} \|\tilde{\Ubf}^{(m),\star}-\Ubf^{(m),\star}\| \leq \tilde{C} m^{-\alpha \min\{\frac{1}{d+1},\frac{k}{d+1}-\frac{1}{2}\}}.\]

Further, this is a worst-case bound reached only when $u$ is both discontinuous and not continuously differentiable in the subregions of $D=\Omega\times(0,T)$ where $u$ is continuous. If $u\in C^{1,\gamma}(D)$ then the rate changes to $\CalO(m^{-\frac{\alpha}{d+1}(1+\gamma)})$.

$I_5$: Asymptotic behavior of $I_5=\|\Gbf^{(m),\star} - \Gbf^\star\|$ is similar to $I_4$ as it is determined by how quickly the trapezoidal rule converges on $\partial^{\alpha^s}\psi(\cdot) f_j(u(\cdot))$. The worst-case rate is given by

\[I_5\leq \tilde{C} m^{-\min\{\frac{1}{d+1},\frac{k}{d+1}-\frac{1}{2}\}},\]

which similary improves with the smoothness of $u$.

$I_3$: For $I_3=\|\widehat{\Gbf}^{(m)} - \overline{\Gbf}^{(m)}\|$, we can consider the pointwise rate of convergence

\[\rho^{(m)}\star f_j(x)\to f_j(x),\]

which for $f_j$ locally Lipschitz and with polynomial growth, we get

\[|\rho^{(m)}\star f_j(x)-f_j(x)|\leq C_\CalF\left(\sqrt{M_2(\rho^{(m)})} + M_{p_{\max}}(\rho^{(m)})\right)\leq C_\CalF C_\sigma\frac{\sigma}{m^{\alpha/2}}\]

where we used Jensen's inequality and sub-Gaussianity of $\rho^{(m)}$ to relate higher moments to the variance, and $C_\sigma$ depends on $p_{max}$ and $\rho$ but not $m$. It should be noted that this rate also increases to $\CalO(m^{-\alpha})$ for polynomial and trigonometric libraries $\CalF$.

Hence for $m$ large enough, it suffices to consider

\[\Pbb\left(I_1+I_2 > t'\right) \leq \Pbb\left(I_1 > t'/2\right)+\Pbb\left(I_2 > t'/2\right) := E_1+E_2,\]

where $t' = t-\sum_{i=3}^5I_i$, and for $k>1+\frac{d+1}{2}$, $t' = t - \tilde{C}m^{-\frac{\alpha}{d+1}}$. For $E_2$ we are in the same situation as in Theorem <ref>, hence for $t'>0$ we can use Corollary <ref> to get the exponential concentration:

\[E_2 = \Pbb\left( \|\widehat{\Gbf}^{(m)} - \overline{\Gbf}^{(m)}\|_{\overrightarrow{\infty}} > t'/2\right) \leq 2K\mathfrak{J}\exp\left(-\frac{c}{2^{1+2/{p_{\max}}}}(mt')^{2/{p_{\max}}}\right).\]

This leaves $E_1$, which we now show also yields exponential concentration, but at a sub-Gaussian rate. Indeed,

\begin{align*}
\Pbb\left(|\tilde{\Gbf}^{(m)}_{ksj} - \widehat{\Gbf}_{ksj}^{(m)}|> t'/2\right) &= \Pbb\left(|\lan \partial^{\alpha^s}\psi_k, f_j(\tilde{\Ubf}^{(m),\star}+\tilde{\ep}^{(m)})-f_j(\tilde{\Ubf}^{(m),\star}+\widehat{\ep}^{(m)})\ran_m| > t'/2\right) \\
&\leq \Pbb\left(\nrm{\partial^{\alpha^s}\psi_k}_\infty\nrm{f_j(\tilde{\Ubf}^{(m),\star}+\tilde{\ep}^{(m)})-f_j(\tilde{\Ubf}^{(m),\star}+\widehat{\ep}^{(m)})}_{\overrightarrow{1}} > mt'/(2\gamma T|\Omega|)\right)\\
&\leq \sum_{i=1}^m \Pbb\left(|f_j(\tilde{\Ubf}^{(m),\star}_i+\tilde{\ep}^{(m)}_i)-f_j(\tilde{\Ubf}^{(m),\star}_i+\widehat{\ep}_i^{(m)})| > Ct'\right)
\end{align*}

where $C = (2\gamma T|\Omega| \nrm{\partial^{\alpha^s}\psi_k}_\infty)^{-1}$. (Here we replaced the volume element $(\Delta x)^d\Delta t$ with the equivalent expression $\gamma\frac{T|\Omega|}{m}$ using that $|\supp{\psi}| = \gamma T|\Omega|$, for some $\gamma<1$.) Now, define the sets

\[A_i = \{|f_j(\tilde{\Ubf}^{(m),\star}_i+\tilde{\ep}^{(m)}_i)-f_j(\tilde{\Ubf}^{(m),\star}_i+\widehat{\ep}^{(m)}_i)| > Ct'\},\]

\[B_i = \{|\tilde{\ep}^{(m)}_i - \widehat{\ep}_i^{(m)}|>1\}.\]

We get

\begin{align*}
\Pbb\left(|f_j(\tilde{\Ubf}^{(m),\star}_i+\tilde{\ep}^{(m)}_i)-f_j(\tilde{\Ubf}^{(m),\star}_i+\widehat{\ep}_i^{(m)})| > Ct'\right)&=\Pbb(A_i\cap B_i)+\Pbb(A_i\cap B_i^c)
\end{align*}

where, using sub-Gaussianity,

\[\Pbb(A_i\cap B_i) \leq \Pbb(B_i)\leq \tilde{C} e^{-\frac{m^\alpha}{2\sigma^2}}\]

and using the locally Lipschitz condition on $f_j$, together with the independent of $\tilde{\ep}^{(m)}_i$ and $\widehat{\ep}^{(m)}_i$,

\begin{align*}
\Pbb(A_i\cap B_i) &\leq \Pbb\left(\{C_\CalF(|\tilde{\ep}^{(m)}_i - \widehat{\ep}_i^{(m)}|+|\tilde{\ep}^{(m)}_i - \widehat{\ep}_i^{(m)}|^{p_{\max}})>Ct'\}\cap\{|\tilde{\ep}^{(m)}_i - \widehat{\ep}_i^{(m)}|<1\}\right)\\
&\leq \Pbb\left(|\tilde{\ep}^{(m)}_i - \widehat{\ep}_i^{(m)}|>\frac{C}{2C_\CalF}t'\right) \\
&\leq \tilde{C} e^{-\frac{m^\alpha}{2\sigma^2}\left(\frac{Ct'}{2C_\CalF}\right)^2}.
\end{align*}

Hence, summing over all $i$, we get

\[\Pbb\left(|\tilde{\Gbf}^{(m)}_{ksj} - \widehat{\Gbf}_{ksj}^{(m)}|> t'/2\right) \leq \tilde{C}me^{-c_1 m^\alpha (t')^2}\]

for some fixed $c_1>0$, which shows that the term $E_2$ asymptotically dominates $E_1$.

Overall this shows that $\tilde{\Gbf}^{(m)}$ converges to $\Gbf^\star$ elementwise, and since the number of elements is fixed, we can take a union bound to conclude that $\|\tilde{\Gbf}^{(m)}-\Gbf^\star\|_\infty\to 0$ in probability at the same rate. We can characterize this convergence as follows. (1) For each $m$ we acrue a deterministic error $\mathfrak{t}(m)$ due to the bias introduced when averaging the clean data, as well as the integration errors. The worst-case asymptotic order of this error is $\CalO(m^{-\alpha \min\{\frac{1}{d+1},\frac{k}{d+1}-\frac{1}{2}\}})$ and arises from $I_4$, the difference between $\overline{\Gbf}^{(m),\star}$ which uses the filtered clean data $\tilde{\Ubf}^{(m),\star}$ and $\Gbf^{(m),\star}$ which uses the clean data $\Ubf^{(m),\star}$ (in effect, $I_4$ is the combined bias and integration error). (2) For $t>\mathfrak{t}(m)$ and large enough $m$ we have

\[\Pbb \left(\|\tilde{\Gbf}^{(m)} - \Gbf^\star\|_\infty>t\right) \leq 3K\mathfrak{J}\exp\left(-\frac{c}{2^{1+2/{p_{\max}}}}[m(t-\mathfrak{t}(m))]^{2/{p_{\max}}}\right),\]

using the worst-case concentration rate arising from the error $I_2$.

§ FULL SUPPORT RECOVERY

As indicated in Remark <ref>, in this section we extend the results (<ref>) of Lemma <ref> and (<ref>) of Theorem <ref> to subset equality

\[\supp{\widehat{\wbf}} = \supp{\wstar} \qquad \& \qquad\supp{\widehat{\wbf}^{(m)}}=\supp{\wstar},\]

respectively, with the addition of a constraint on the linear system $(\Gbf^\star,\bbf^\star)$. First we need another stability lemma similar to Lemma <ref>.

Let $\Abf \in \Rbb^{m\times n}$ have rank $n$ and let $\tilde{\Abf} = \Abf+\Ebf$ be a perturbed system satisfying $\nrm{\Ebf}_{2}<\varepsilon$. For sufficiently small $\varepsilon$, there exist constants $C,C'>0$ depending only on $\Abf$ such that the following stability holds for the pseudoinverse $\Abf^\dagger := (\Abf^T\Abf)^{-1}\Abf^T$ and projection operator $\Pbf_\Abf := \Abf\Abf^\dagger$:

\begin{equation}
\nrm{\Abf^\dagger - \tilde{\Abf}^\dagger}_2<C\varepsilon \qquad \& \qquad \nrm{\Pbf_\Abf-\Pbf_{\tilde{\Abf}}}_2<C'\varepsilon.
\end{equation}

The pseudoinverse of $\tilde{\Abf}$ is given by

\[\tilde{\Abf}^\dagger = (\tilde{\Abf}^T\tilde{\Abf})^{-1}\tilde{\Abf}^T = (\Ibf+\Bbf)^{-1}(\Abf^\dagger+(\Abf^T\Abf)^{-1}\Ebf^T),\]

provided $(\Ibf+\Bbf)^{-1}$ exists, where

\[\Bbf = \Abf^\dagger\Ebf+(\Abf^T\Abf)^{-1}\Ebf^T(\Abf + \Ebf).\]

A sufficient condition for invertibility of $\Ibf+\Bbf$ is

\[\nrm{\Bbf}_2<1,\]

which is guaranteed for sufficiently small $\varepsilon$ since $\nrm{\Bbf}_2 = \CalO(\varepsilon)$. In this case using the Von Neumann series for $(\Ibf+\Bbf)^{-1}$, we get that

\[\tilde{\Abf}^\dagger = \Abf^\dagger + \tilde{\Ebf},\]

\[\tilde{\Ebf} = (\Abf^T\Abf)^{-1}\Ebf^T+\sum_{k=1}^\infty(-\Bbf)^k\left(\Abf^\dagger+(\Abf^T\Abf)^{-1}\Ebf^T\right).\]

Since $\Ebf=\CalO(\varepsilon)$ and $\Bbf = \CalO(\varepsilon)$, we have $\tilde{\Ebf} = \CalO(\varepsilon)$, hence

\[\nrm{\Abf^\dagger - \tilde{\Abf}^\dagger}_2 = \nrm{\tilde{\Ebf}}_2 = \CalO(\varepsilon).\]

The same stability readily applies to $\Pbf_{\tilde{\Abf}} = \Pbf_{\Abf} + \Abf\tilde{\Ebf}+\Ebf\tilde{\Abf}^\dagger$.

We can now use this stability to extend Lemma <ref> to yield full support recovery on the continuum problem for small enough $\sigma^2$, provided condition (<ref>) is satisfied.

Let $\rho$ be a Gaussian mean-zero noise distribution and let $p$ be the maximum polynomial degree appearing in the true model. Assume that $(\Gbf^\star,\bbf^\star)$ satisfy

\begin{equation}\label{app:condforsupprec}
\mu^\star:= \min_{S\subsetneq S^\star}\frac{\nrm{\Pbf^\perp_{\Gbf^\star_{S^\star\setminus S}} \bbf^\star}}{\nrm{\bbf^\star}}-\frac{|S|+1}{\mathfrak{J}}>0.
\end{equation}

Then there exists a critical noise level $\sigma_c$ such that for any $\sigma\leq \sigma_c$ the estimator $\what = \text{MSTLS}^{(1)}(\overline{\Gbf},\overline{\bbf})$ satisfies

\begin{equation}\label{subsetsupprec_lemm_app}
\supp{\what}=\supp{\wstar}.
\end{equation}

In Lemma <ref> we already showed that

* For sufficiently small $\sigma$ there exists $\widehat{\lambda}\in \pmb{\lambda}$ such that $\supp{\overline{\wbf}^{\widehat{\lambda}}} = \supp{\wstar}$, where $\overline{\wbf}^{\widehat{\lambda}} = \text{STLS}^{(1)}(\overline{\Gbf},\overline{\bbf},\widehat{\lambda})$.

* For sufficiently small $\sigma$ we have

\[\frac{\nrm{\overline{\Gbf}\left(\overline{\wbf}^0 -\overline{\wbf}^{\widehat{\lambda}}\right)}_2}{\nrm{\overline{\bbf}}_2} <\frac{1}{\mathfrak{J}}\]

which implies that for $\lambda\in \pmb{\lambda}$ with $\lambda<\widehat{\lambda}$, we have

\[\CalL(\widehat{\lambda})<\CalL(\lambda).\]

Assume that $\sigma$ is small enough so that (a) and (b) hold, and consider $\lambda\in \pmb{\lambda}$ with $\lambda>\widehat{\lambda}$. All we need to show is that for sufficiently small $\sigma$, we have $\CalL(\widehat{\lambda})<\CalL(\lambda)$ in this case as well. Indeed, by construction of $\pmb{\lambda}$, for $\lambda>\widehat{\lambda}$ we have

\[\supp{\overline{\wbf}^{\lambda}} \subsetneq S^\star.\]

Setting $S = S^\star \setminus \supp{\overline{\wbf}^{\lambda}}$, we see that

\[\CalL(\widehat{\lambda}) - \CalL(\lambda) = \frac{\nrm{\overline{\Gbf}\left(\overline{\wbf}^0 -\overline{\wbf}^{\widehat{\lambda}}\right)}_2}{\nrm{\overline{\bbf}}_2} - \frac{\nrm{\overline{\Gbf}\left(\overline{\wbf}^0 -\overline{\wbf}^{\lambda}\right)}_2}{\nrm{\overline{\bbf}}_2}+\frac{|S|}{\mathfrak{J}}.\]

Using (b), and rewriting using the projection operator, we have

\[\CalL(\widehat{\lambda}) - \CalL(\lambda) < - \frac{\nrm{\overline{\Gbf}\left(\overline{\wbf}^0 -\overline{\wbf}^{\lambda}\right)}_2}{\nrm{\overline{\bbf}}_2}+\frac{|S|+1}{\mathfrak{J}} = -\frac{\nrm{\Pbf^\perp_{\overline{\Gbf}_{S^\star \setminus S}}\overline{\bbf}}_2}{\nrm{\overline{\bbf}}_2}+\frac{|S|+1}{\mathfrak{J}}.\]

Now, for small enough $\sigma$, using stability of the projection operator, we have

\[\max_{S\subsetneq S^\star}\nrm{\Pbf^\perp_{\overline{\Gbf}_{S^\star \setminus S}} - \Pbf^\perp_{\Gbf^\star_{S^\star \setminus S}} }_2 < C\sigma^2,\]

since properties of $\Dbf$ and $\Lbf$ imply

\[\overline{\Gbf}_{S^\star \setminus S} -\Gbf^\star_{S^\star \setminus S} = \Gbf^\star (\Abf_{S^\star \setminus S} -\Ibf_{S^\star \setminus S}) = \Gbf^\star(\Dbf-\Ibf)_{S^\star \setminus S}+ \Gbf^\star \Lbf_{S^\star \setminus S} =  \CalO(\sigma^2).\]

Hence, using the reverse triangle inequality we get

\begin{align*}
\CalL(\widehat{\lambda}) - \CalL(\lambda) &< -\frac{\nrm{\Pbf^\perp_{\Gbf^\star_{S^\star \setminus S}}\overline{\bbf}}_2}{\nrm{\overline{\bbf}}_2} + \frac{\nrm{\left(\Pbf^\perp_{\Gbf^\star_{S^\star \setminus S}}- \Pbf^\perp_{\Gbf^\star_{S^\star \setminus S}}\right)\overline{\bbf}}_2}{\nrm{\overline{\bbf}}_2} +\frac{|S|+1}{\mathfrak{J}} \\
&< -\mu^\star + C\sigma^2
\end{align*}

which is negative for all $\sigma^2 <\mu^\star/C$. Thus, to recover the full support on systems satisfying (<ref>), in addition to other constraints derived on $\sigma_c$, it holds that $\sigma_c^2\geq \mu^\star/C$.

The previous Lemma, together with results shown in Section <ref>, directly imply the following, using similar techniques as Theorem <ref>.

Let Assumptions <ref>-<ref> hold with $\rho$ a mean-zero Gaussian distribution. In addition assume that the condition (<ref>) holds. Then exists a critical noise $\sigma_c>0$ and a stability tolerance $\tau$, both independent of $m$, such that for all $\sigma<\sigma_c$ and $t< \tau$, and for sufficiently large $m$, it holds that

\begin{equation}\label{supprec_thm}
\supp{\widehat{\wbf}^{(m)}}=\supp{\wstar} \qquad \text{and} \qquad \nrm{\widehat{\wbf}^{(m)}-\wstar}_\infty < C'(t+\sigma^2)
\end{equation}

with probability exceeding $1-2K\mathfrak{J}\exp\left(-\frac{c}{2}\left(mt\right)^{2/p_{\max}}\right)$, where $\widehat{\wbf}^{(m)} = \text{MSTLS}^{(1)}(\Gbf^{(m)},\bbf^{(m)})$, $c$ is from Theorem <ref>, and $C'$ depends on only $(\Gbf^\star_{S^\star},\bbf^\star)$.

§ HYPER-KS DYNAMICS

\begin{equation}\label{hyperKSapp}
\partial_t u = (1)\partial_{xxxx} u + (0.75)\partial_{xxxxxx} u + (-0.5)\partial_x(u^2) + (0.1)\partial_{xxx} u^2.
\end{equation}

Motivation for equation <ref> comes from the fact that variants of this equation arise when performing equation learning on corrupted Kuramoto-Sivashinsky data using a disadvantageous test function, if derivatives up to order 6 are included in the library. (See for example Figure 16 of [32].) Two effects dominate the dynamics. At low wavelengths $|k|\leq k^*$ below some threshold $k^*$ the system is unstable. Then at all wavelengths sufficiently separated from a critical wavenumber $\overline{k}$, we have dispersive mixing of wavemodes. At the critical wavenumber $\overline{k}$, dispersive and transport effects cancel out, such that $\overline{k}$ is entirely subject to the growth and decay dynamics of the diffusive terms. This implies that if $|\overline{k}|< k^*$, the system can become unstable.

§ VARIANCE ESTIMATION

We estimate the noise variance using the following procedure. Let $\fbf\in \Rbb^{2L+1}$ satisfy the discrete moment conditions

\[M_k(\fbf) = \sum_{j=-L}^L  j^k\fbf_f = 0, \quad k=0,\dots,M_{\max}.\]

\[\nrm{\fbf}_2=1.\]

Now consider data $\Ubf = u(\tbf)+\ep$ such that $u$ is locally a polynomial of degree $M_{\max}$ over intervals $(t-L\Delta t, t+L\Delta t)$ and $\ep$ is mean zero i.i.d. noise with variance $\sigma^2$. Then the data $\Ubf$ will satisfy

\[\Ebb\left[|(\fbf*\Ubf)^2_i\right] = \sigma^2, \quad i\in \Zbb,\]

\[\nrm{\fbf*\Ubf}_{rms}\approx \sigma\]

to very high accuracy. We use this approach and take $\fbf$ to be 6th-order finite difference weights normalized so that $\nrm{\fbf}_2=1$.

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