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Convergence and error control of consistent PINNs for elliptic PDEs

Andrea Bonito, Ronald DeVore, Guergana Petrova, Jonathan W. Siegel

TL;DR

The paper develops a rigorous, Besov-space–based framework for analyzing collocation methods solving elliptic PDEs, including Physics-Informed Neural Networks (PINNs). It derives matching upper and lower bounds on optimal recovery rates in the $H^1(\Omega)$ norm for right-hand sides and boundary data with Besov smoothness, and shows how to discretize norms to form a computable, data-driven loss ${\cal L}^*$ that yields an a priori error bound for near-minimizers. A key contribution is the design of consistent PINNs via ${\cal L}^*$, and proofs that minimization over sufficiently expressive neural spaces can achieve near-optimal recovery provided approximation properties hold. Numerical experiments in 2D Poisson problems illustrate that consistent PINNs with the new loss functions improve solution accuracy, particularly for low-regularity data, and provide practical guidance on loss design and boundary handling. The work thus connects optimal recovery theory, Besov regularity, and NN-based PDE solvers, offering certified error control and a principled path to reliable PINN implementations for elliptic problems.

Abstract

We provide an a priori analysis of collocation methods for solving elliptic boundary value problems. They begin with information in the form of point values of the data and utilize only this information to numerically approximate the solution u of the PDE. For such a method to provide an approximation with guaranteed error bounds, additional assumptions on the data, called model class assumptions, are needed. We determine the best error of approximating u in the energy norm, in terms of the total number of point samples, under all Besov class model assumptions for the right hand side and boundary data. We then turn to the study of numerical procedures and analyze whether a proposed numerical procedure achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to $u$ by minimizing specified data driven loss functions over a set $Σ$ which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a new loss function is proposed. We also address the recent methods of Physics Informed Neural Networks. We prove that minimization of the new loss over restricted neural network spaces $Σ$ provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $Σ$ has sufficient approximation capabilities. We also analyze variants of the new loss function which are more practical for implementation. Finally, numerical examples illustrating the benefits of the proposed loss functions are given.

Convergence and error control of consistent PINNs for elliptic PDEs

TL;DR

The paper develops a rigorous, Besov-space–based framework for analyzing collocation methods solving elliptic PDEs, including Physics-Informed Neural Networks (PINNs). It derives matching upper and lower bounds on optimal recovery rates in the norm for right-hand sides and boundary data with Besov smoothness, and shows how to discretize norms to form a computable, data-driven loss that yields an a priori error bound for near-minimizers. A key contribution is the design of consistent PINNs via , and proofs that minimization over sufficiently expressive neural spaces can achieve near-optimal recovery provided approximation properties hold. Numerical experiments in 2D Poisson problems illustrate that consistent PINNs with the new loss functions improve solution accuracy, particularly for low-regularity data, and provide practical guidance on loss design and boundary handling. The work thus connects optimal recovery theory, Besov regularity, and NN-based PDE solvers, offering certified error control and a principled path to reliable PINN implementations for elliptic problems.

Abstract

We provide an a priori analysis of collocation methods for solving elliptic boundary value problems. They begin with information in the form of point values of the data and utilize only this information to numerically approximate the solution u of the PDE. For such a method to provide an approximation with guaranteed error bounds, additional assumptions on the data, called model class assumptions, are needed. We determine the best error of approximating u in the energy norm, in terms of the total number of point samples, under all Besov class model assumptions for the right hand side and boundary data. We then turn to the study of numerical procedures and analyze whether a proposed numerical procedure achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to by minimizing specified data driven loss functions over a set which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a new loss function is proposed. We also address the recent methods of Physics Informed Neural Networks. We prove that minimization of the new loss over restricted neural network spaces provides an optimal recovery of the solution , provided that the optimization problem can be numerically executed and has sufficient approximation capabilities. We also analyze variants of the new loss function which are more practical for implementation. Finally, numerical examples illustrating the benefits of the proposed loss functions are given.
Paper Structure (39 sections, 17 theorems, 254 equations, 3 figures, 3 tables)

This paper contains 39 sections, 17 theorems, 254 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Numerical methods for solving (\ref{['main']})
  3. Collocation methods and PINNs
  4. Overview of this paper
  5. Besov spaces
  6. The Sobolev spaces $H^s(\Omega)$
  7. Piecewise polynomial approximation and interpolation
  8. Optimal recovery
  9. Optimal recovery of $f$
  10. Optimal recovery of $g$
  11. Optimal recovery of $u$
  12. Final observations on optimal recovery
  13. Numerical algorithms based on linear approximation
  14. Using FEMs or AFEMs
  15. Reduced models
  16. ...and 24 more sections

Key Result

Theorem 2.1

Let $s>0$, $0<p\leq \infty$, be fixed with $s>d/p$. Let $r=r(s)$ be the smallest integer strictly larger than $\max(s,1)$ and let $S_k^*$ be the interpolation operator (pwpinter1). Then, for any $f\in B_p^s(\Omega)=B_\infty^s(L_p(\Omega))$ and any $\tau\ge p$, we have with $C$ independent of $f$ and $k$.

Figures (3)

  • Figure 3.1: The spaces $B^{s}_\infty(L_p(\Omega))$ are represented by the point $(1/p,s)$ in the first quadrant. The arrows indicate embedding with a slope of $d$ for the oblique lines. In particular, the spaces above the line $s=d/p$ embed into $C(\overline{\Omega})$. (Left) The largest model classes for $f$ are the unit balls of spaces just above the thick and dashed segment or on the on the thick vertical half-line (excluding the point $(1/{\delta},d/{\delta})$). (Right) The largest model classes for $g$ are the traces of the unit balls of spaces on the thick vertical half-line (excluding the point $(1/2,d/2)$).
  • Figure 10.2: Experiment 3: Solution plots for the (unweighted) original PINNs loss ${\cal L}_{sq,1}$ (left) and the consistent PINNs loss ${\cal L}^*_{sq,\gamma}$ (right). We see that both fail to accurately fit the boundary values. Note that the consistent PINNs solution is much less smooth near the boundary.
  • Figure 10.3: Experiment 3: Solution plots for the correctly weighted PINNs loss ${\cal L}_{sq,\lambda(\overline{m})}$ (left) and the consistent PINNs loss with $L_2$-penalty on the domain ${\cal L}^*_{sq,2}$ (right). We see that both accurately fit the boundary values and remain fairly smooth.

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Lemma 4.1
  • ...and 29 more