# Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

arXiv: https://arxiv.org/abs/2402.10926 · HTML: https://mlsanity.com/paper/2402.10926/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 108 references that are in the map.
- First posted on arXiv 30 Jan 2024.

## Builds on (most cited first, 23 of 23)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Parametric Complexity Bounds for Approximating PDEs with Neural Networks](https://mlsanity.com/paper/2103.02138/) (2021)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)

## Built on by (most cited first, 30 of 39)

- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [Operator Learning at Machine Precision](https://mlsanity.com/paper/2511.19980/) (2025)
- [First Order System Least Squares Neural Networks](https://mlsanity.com/paper/2409.20264/) (2024)
- [A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.13717/) (2025)
- [Aspects of holographic entanglement using physics-informed-neural-networks](https://mlsanity.com/paper/2509.25311/) (2025)
- [Prediction error certification for PINNs: Theory, computation, and application to Stokes flow](https://mlsanity.com/paper/2508.07994/) (2025)
- [Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs](https://mlsanity.com/paper/2508.00628/) (2025)
- [Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm](https://mlsanity.com/paper/2602.05052/) (2026)
- [HyperNOs: Automated and Parallel Library for Neural Operators Research](https://mlsanity.com/paper/2503.18087/) (2025)
- [Noisy PDE Training Requires Bigger PINNs](https://mlsanity.com/paper/2507.06967/) (2025)
- [Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equation](https://mlsanity.com/paper/2508.21571/) (2025)
- [Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation](https://mlsanity.com/paper/2509.09362/) (2025)
- [A data augmentation strategy for deep neural networks with application to epidemic modelling](https://mlsanity.com/paper/2502.21033/) (2025)
- [Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Data](https://mlsanity.com/paper/2503.10428/) (2025)
- [Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra](https://mlsanity.com/paper/2505.11638/) (2025)
- [Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions](https://mlsanity.com/paper/2508.07559/) (2025)
- [Augmented data and neural networks for robust epidemic forecasting: application to COVID-19 in Italy](https://mlsanity.com/paper/2510.09192/) (2025)
- [Physically consistent model learning for reaction-diffusion systems](https://mlsanity.com/paper/2512.14240/) (2025)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [THINNs: Thermodynamically Informed Neural Networks](https://mlsanity.com/paper/2509.19467/) (2025)
- [Domain-Informed Genetic Superposition Programming: A Case Study on SFRC Beams](https://mlsanity.com/paper/2509.21355/) (2025)
- [Rethinking Physics-Informed Regression Beyond Training Loops and Bespoke Architectures](https://mlsanity.com/paper/2512.13217/) (2025)
- [Gauss-Newton Natural Gradient Descent for Shape Learning](https://mlsanity.com/paper/2602.00099/) (2026)
- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)
- [Building Trust in PINNs: Error Estimation through Finite Difference Methods](https://mlsanity.com/paper/2603.15526/) (2026)
- [A Family of Adaptive Activation Functions for Mitigating Failure Modes in Physics-Informed Neural Networks](https://mlsanity.com/paper/2603.18328/) (2026)
- [A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networks](https://mlsanity.com/paper/2604.04971/) (2026)
- [DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting](https://mlsanity.com/paper/2604.18261/) (2026)
- [Random Neural Network Expressivity for Non-Linear Partial Differential Equations](https://mlsanity.com/paper/2605.25057/) (2026)
- [Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations](https://mlsanity.com/paper/2606.03936/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified framework in which analysis of the various components of the error incurred by PINNs in approximating PDEs can be effectively carried out. A detailed review of available results on approximation, generalization and training errors and their behavior with respect to the type of the PDE and the dimension of the underlying domain is presented. In particular, the role of the regularity of the solutions and their stability to perturbations in the error analysis is elucidated. Numerical results are also presented to illustrate the theory. We identify training errors as a key bottleneck which can adversely affect the overall performance of various models in physics-informed machine learning.