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Efficient Error Certification for Physics-Informed Neural Networks

Francisco Eiras, Adel Bibi, Rudy Bunel, Krishnamurthy Dj Dvijotham, Philip Torr, M. Pawan Kumar

TL;DR

This work addresses the absence of global, continuous-domain error guarantees for Physics-Informed Neural Networks (PINNs) solving PDEs by introducing ∂-CROWN, a post-training framework that certifies PINN residuals and boundary compliance over the entire domain. It combines CROWN-style linear relaxations with new bounds on partial derivatives, enabling δ0, δb, and ε-type correctness guarantees for initial, boundary, and residual errors. The framework is demonstrated on Burgers', Schrödinger's, Allan–Cahn, and Diffusion-Sorption equations, yielding tight certified bounds that closely match empirical estimates and revealing a practical link between residual magnitude and true solution error. Greedy input branching further tightens bounds by adaptively partitioning the input domain, offering scalable certification despite the computational challenges of higher-order PDEs and larger PINNs, with implications for safer deployment and improved training strategies.

Abstract

Recent work provides promising evidence that Physics-Informed Neural Networks (PINN) can efficiently solve partial differential equations (PDE). However, previous works have failed to provide guarantees on the worst-case residual error of a PINN across the spatio-temporal domain - a measure akin to the tolerance of numerical solvers - focusing instead on point-wise comparisons between their solution and the ones obtained by a solver on a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment, as the performance could be substantially worse on a different set. To alleviate this issue, we establish guaranteed error-based conditions for PINNs over their continuous applicability domain. To verify the extent to which they hold, we introduce $\partial$-CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classically studied PINNs - Burgers' and Schrödinger's equations -, and two more challenging ones with real-world applications - the Allan-Cahn and Diffusion-Sorption equations.

Efficient Error Certification for Physics-Informed Neural Networks

TL;DR

This work addresses the absence of global, continuous-domain error guarantees for Physics-Informed Neural Networks (PINNs) solving PDEs by introducing ∂-CROWN, a post-training framework that certifies PINN residuals and boundary compliance over the entire domain. It combines CROWN-style linear relaxations with new bounds on partial derivatives, enabling δ0, δb, and ε-type correctness guarantees for initial, boundary, and residual errors. The framework is demonstrated on Burgers', Schrödinger's, Allan–Cahn, and Diffusion-Sorption equations, yielding tight certified bounds that closely match empirical estimates and revealing a practical link between residual magnitude and true solution error. Greedy input branching further tightens bounds by adaptively partitioning the input domain, offering scalable certification despite the computational challenges of higher-order PDEs and larger PINNs, with implications for safer deployment and improved training strategies.

Abstract

Recent work provides promising evidence that Physics-Informed Neural Networks (PINN) can efficiently solve partial differential equations (PDE). However, previous works have failed to provide guarantees on the worst-case residual error of a PINN across the spatio-temporal domain - a measure akin to the tolerance of numerical solvers - focusing instead on point-wise comparisons between their solution and the ones obtained by a solver on a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment, as the performance could be substantially worse on a different set. To alleviate this issue, we establish guaranteed error-based conditions for PINNs over their continuous applicability domain. To verify the extent to which they hold, we introduce -CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classically studied PINNs - Burgers' and Schrödinger's equations -, and two more challenging ones with real-world applications - the Allan-Cahn and Diffusion-Sorption equations.
Paper Structure (46 sections, 5 theorems, 72 equations, 11 figures, 8 tables, 1 algorithm)

This paper contains 46 sections, 5 theorems, 72 equations, 11 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Physics-informed Neural Networks.
  4. Robustness Verification of Neural Networks.
  5. Preliminaries
  6. Notation
  7. Physics-informed neural networks (PINNs)
  8. Bounding neural network outputs using CROWN (Zheng et al., 2018
  9. ∂-CROWN: Error Certification for General Physics-Informed Neural Networks
  10. Bounding Partial Derivatives of u_θ
  11. Bounding f_θ
  12. Tighter Bounds via Greedy Input Branching
  13. Experiments
  14. Certifying with ∂-CROWN
  15. Burgers' Equation
  16. ...and 31 more sections

Key Result

Lemma 1

For $i\in\{1,\dots,d_0\}$, the partial derivative of $u_\theta$ with respect to $\mathbf{x}_i$ can be computed recursively as $\partial_{\mathbf{x}_i} u_\theta = \mathbf{W}^{(L)} \partial_{\mathbf{x}_i} z^{(L-1)}$ for: for $k \in \{1,\dots,L-1\}$, and where $\partial_{z^{(k-1)}} z^{(k)} = \text{diag}\left[\sigma'\left(y^{(k)}\right)\right]\mathbf{W}{^{(k)}}$.

Figures (11)

  • Figure 1: Bounding Partial Derivatives with $\partial$-CROWN: our hybrid scheme for bounding $\partial_{\mathbf{x}_i} u_\theta$ and $\partial_{\mathbf{x}_i^2} u_\theta$ uses back-propagation and forward substitution (inspired by shi2020robustness) to compute bounds in $\mathcal{O}(L)$ instead of the $\mathcal{O}(L^2)$ complexity of full back-propagation as in xu2020automatic.
  • Figure 2: Residual and solution errors: connection of the maximum residual error ($\max_{\mathcal{S}'} |f_\theta|$) and the maximum solution error, $\max_{\mathcal{S}'} |u_\theta - \tilde{u}|$, for networks at different epochs of the training process (in orange).
  • Figure 3: Branching densities: relative density of the input branching distribution obtained via Algorithm \ref{['alg:greedy_input_branching']} applied to Burgers' (top) and Schrödinger's (bottom) equations.
  • Figure 4: Certifying with $\partial$-CROWN: visualization of the time evolution of $u_\theta$, and the residual errors as a function of the spatial temporal domain (log-scale), $|f_\theta|$, for (a) Burgers' equation raissi2019physics, (b) Schrödinger's equation raissi2019physics, (c) Allan-Cahn's equation monaco2023training, and (d) the Diffusion-Sorption equation takamoto2022pdebench.
  • Figure 5: Certification Convergence: log-log plot of the relative convergence of $\partial$-CROWN certification for a standard trained PINN (in blue) and PIAT (in orange).
  • ...and 6 more figures

Theorems & Definitions (7)

  • Definition 1: Correctness Conditions for PINNs
  • Lemma 1: Expression for $\partial_{\mathbf{x}_i} u_\theta$
  • Theorem 1: Informal, $\partial$-CROWN Linear Bounding $\partial_{\mathbf{x}_i} u_\theta$
  • Lemma 2: Expression for $\partial_{\mathbf{x}_i^2} u_\theta(\mathbf{x})$
  • Theorem 2: Informal, $\partial$-CROWN Linear Bounding $\partial_{\mathbf{x}_i^2} u_\theta$
  • Lemma 3: Closed-form global bounds on $\partial_{\mathbf{x}_i} u_\theta$
  • proof