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A Unified Framework for the Error Analysis of Physics-Informed Neural Networks

Marius Zeinhofer, Rami Masri, Kent-André Mardal

TL;DR

This work develops a unified, coercivity based error analysis for physics-informed neural networks (PINNs) solving linear PDEs across elliptic, parabolic, hyperbolic, Stokes, and PDE constrained optimization problems. It introduces an abstract least-squares energy framework and a non-Galerkin Céa type analysis that yields both a priori and a posteriori error estimates by combining energy, duality, and interpolation arguments. A key finding is that $L^2$ penalties for boundary and initial data can degrade the decay rate of the error in natural Sobolev norms, while encoding constraints directly into the neural representation avoids this issue. The authors provide sharp coercivity and continuity results for Poisson, Darcy, elasticity, Stokes, parabolic, hyperbolic, and source recovery problems and corroborate the theory with three- to four-dimensional numerical experiments using recent optimization advances. Together these results offer rigorous guarantees and practical guidance for designing PINNs that solve linear PDEs efficiently with provable accuracy.

Abstract

We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^2$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.

A Unified Framework for the Error Analysis of Physics-Informed Neural Networks

TL;DR

This work develops a unified, coercivity based error analysis for physics-informed neural networks (PINNs) solving linear PDEs across elliptic, parabolic, hyperbolic, Stokes, and PDE constrained optimization problems. It introduces an abstract least-squares energy framework and a non-Galerkin Céa type analysis that yields both a priori and a posteriori error estimates by combining energy, duality, and interpolation arguments. A key finding is that penalties for boundary and initial data can degrade the decay rate of the error in natural Sobolev norms, while encoding constraints directly into the neural representation avoids this issue. The authors provide sharp coercivity and continuity results for Poisson, Darcy, elasticity, Stokes, parabolic, hyperbolic, and source recovery problems and corroborate the theory with three- to four-dimensional numerical experiments using recent optimization advances. Together these results offer rigorous guarantees and practical guidance for designing PINNs that solve linear PDEs efficiently with provable accuracy.

Abstract

We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.
Paper Structure (28 sections, 25 theorems, 174 equations, 5 tables)

This paper contains 28 sections, 25 theorems, 174 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Existing Literature
  3. Main Contributions
  4. Proof Strategy
  5. Preliminaries
  6. Abstract Framework
  7. Linear PDEs
  8. Approximation results
  9. Applications of the theory
  10. Poisson's Equation
  11. Darcy's Equations
  12. Linear Elasticity
  13. Stokes System
  14. Parabolic Equations
  15. Second-Order Hyperbolic Equations
  16. ...and 13 more sections

Key Result

Theorem 1

Assume we are in Setting setting:linear_equations and Assumption (A1) holds. Let $\mathcal{F}_\Theta$ be a subset of $X$. For an arbitrary element $u_{\theta^*}\in \mathcal{F}_\Theta$ we can estimate where $\eta(\theta^*)$ represents a local quadrature error $\eta(\theta^*) = E(u_{\theta^*}) - L(\theta^*)$. If (A3) holds, then we can estimate $\eta(\theta^*) \leq \eta.$

Theorems & Definitions (63)

  • Example 1: Forward Problem
  • Example 2: Inverse Problem
  • Remark 1: Validity of the assumptions
  • Remark 2: Bounds on the quadrature error
  • Theorem 1: A Posteriori Error Control
  • proof
  • Theorem 2: Céa's Lemma
  • proof
  • Theorem 3
  • Theorem 4
  • ...and 53 more