Building Trust in PINNs: Error Estimation through Finite Difference Methods

Aleksander Krasowski; René P. Klausen; Aycan Celik; Sebastian Lapuschkin; Wojciech Samek; Jonas Naujoks

cs.LGMar 16, 2026

Building Trust in PINNs: Error Estimation through Finite Difference Methods

Authors: Aleksander Krasowski, René P. Klausen, Aycan Celik, Sebastian Lapuschkin, Wojciech Samek, Jonas Naujoks

Organizations: Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute · Centre of eXplainable Artificial Intelligence, Technological University Dublin · Department of Electrical Engineering and Computer Science, Technische Universität Berlin · BIFOLD - Berlin Institute for the Foundations of Learning and Data

Abstract

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

Figures & tables

Problem	PDE	Boundary Conditions
Poisson	$\nabla^{2}u-f=0$	Dirichlet: $u(x_{L})=u(x_{R})=0$
Heat	$\frac{\partial u}{\partial t}-\alpha\tfrac{\partial^{2}u}{\partial x^{2}}=0$	Dirichlet: $u(x_{L})=u(x_{R})=0$
Drift-Diffusion	$\tfrac{\partial u}{\partial t}-\alpha\tfrac{\partial^{2}u}{\partial x^{2}}-\beta\tfrac{\partial u}{\partial x}=0$	Periodic: $u(x_{L})=u(x_{R})$
Wave	$\tfrac{\partial^{2}u}{\partial t^{2}}-c^{2}\tfrac{\partial^{2}u}{\partial x^{2}}=0$	Dirichlet: $u(x_{L})=u(x_{R})=0$

Table 1: Benchmark problems. The initial condition is

u(x,0)=\sin(2\pi x)

for all time-dependent problems. Parameters are:

\alpha=\tfrac{1}{20}

\beta=2

c=\frac{1}{2}

Panel of Figure 1 — Figure 1: Exact solutions for the benchmark problems. The horizontal line in (a) at $y=0.5$ represents the solution for the 1D Poisson problem.

Panel of Figure 2 — Figure 2: Error estimates for a well-trained PINN on the heat equation, (a)–(d) at four different time-points across space and (e) $L_{2}$ error over time, including the bound $e_{\mathrm{bound}}$ . The FDM-based methods use a $64\times 64$ spatio-temporal grid (time slices shown at nearest grid point to specified values); $e_{\mathrm{bound}}$ uses $64$ spatial points with adaptive time integration (up to $165$ steps at $t=1$ ).

Panel of Figure 3 — Figure 3: Accuracies of the error estimates across different discretizations, measured as $\left\lVert e_{\mathrm{true}}-e_{\mathrm{method}}\right\rVert_{2}$ on respective grid points, for all benchmark problems. Averaged over 10 runs, bands show one standard deviation.

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

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Civil and Environmental Engineering Department University of Illinois Urbana-Champaign Urbana, IL 61801, USA · Pacific Northwest National Laboratory Richland, WA 99352, USA

Jun 10, 2026J/K move · Enter open · S save

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds

Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov

Physikalisch-Technische Bundesanstalt (PTB), Germany · Technical University of Berlin, Germany · Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany+1

Dynamical Systems Neural Network Verification Ordinary Differential Equations Physics-Informed ML

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Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous a posteriori upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two-sided error enclosures. In this paper, we derive computable a posteriori lower bounds for PINN errors in ordinary differential equations on suitable certified state-space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural-network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time-invariant and time-varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed-residual finite-probe certificate based on coordinate unit vectors. We also formulate a certificate-informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post-training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.

Abstract

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