Full error analysis for deep learning PDE solvers

Establish a complete error analysis for deep learning-based approximation schemes for partial differential equations (including physics-informed neural networks, deep Galerkin methods, deep BSDE methods, and the deep Kolmogorov method) by rigorously bounding the overall approximation error between the exact PDE solution and the neural network realization, accounting simultaneously for approximation, sampling/generalization, and optimization errors under reasonable assumptions.

Background

The paper develops an error analysis for the deep Kolmogorov method applied to heat equations, providing bounds that decompose the total error into contributions from architecture size, sample size in the loss function, and optimization error. While classical numerical methods for PDEs, such as finite differences and finite elements, have well-established error analyses, analogous comprehensive results for deep learning methods remain limited.

The authors emphasize that, beyond partial results for specific algorithms and settings, a general and complete theory that rigorously quantifies the overall error for deep learning solvers of PDEs is still lacking. Their work advances the state of the art for the deep Kolmogorov method, but a unified full error analysis for broad classes of deep learning PDE solvers is identified as an open research problem.

References

it basically remains a fundamental open problem of research to establish a full error analysis for any reasonable deep learning approximation scheme for PDEs.

— Error analysis for the deep Kolmogorov method  (2508.17167 - Cîmpean et al., 23 Aug 2025) in Introduction (Section 1)

Finally, a fundamental open problem concerns the quantitative relationship between the size of the training dataset and the optimization landscape of physics-informed neural networks. In particular, it would be of great interest to derive estimates on the number of points required to ensure that the empirical loss is sufficiently close to its continuous counterpart and that the training dynamics effectively reaches a regime where the loss is fully minimized.

— Lazy training of quantum physics informed neural networks  (2609.19239 - Hernandez et al., 16 Sep 2026) in Section 6, Conclusions

However, quantitative error estimates and rigorous convergence guarantees for degenerate elliptic operators such as the $p$-Laplacian and the infinity Laplacian are generally unavailable.

— Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems  (2609.30809 - Yeung et al., 25 Sep 2026) in Section 5, DeepONet introduction, immediately before Section 5.1

The contrast between these relatively small local operator-action errors and the much larger global modeling error $E_{\mathrm{model}}$ in Table~\ref{tab:high-contrast-accuracy} indicates that the high-contrast problem is sensitive to perturbations of the local DtN operators. In particular, average accuracy on the present probe families does not by itself guarantee accuracy of the assembled global solution. This may reflect the conditioning of the global skeleton system or the presence of coefficient-dependent trace directions that are especially important in the high-contrast regime. A systematic analysis of this error amplification, together with the use of coefficient-adapted coarse trace spaces, is left for future work.

— A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media  (2608.25850 - Haines et al., 26 Aug 2026) in Section 6, subsection “Two-dimensional high-contrast experiments”

No asymptotic convergence has been established, whatever the case.

— Comparison of neural and spline representations for physics-informed learning  (2610.06294 - Duvigneau, 5 Oct 2026) in Section Discussion