# Neural network augmented inverse problems for PDEs

arXiv: https://arxiv.org/abs/1712.09685 · HTML: https://mlsanity.com/paper/1712.09685/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2017.

## Built on by (most cited first, 11 of 11)

- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)
- [PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence](https://mlsanity.com/paper/2105.08633/) (2021)
- [TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems](https://mlsanity.com/paper/2105.12033/) (2021)
- [Short-term traffic prediction using physics-aware neural networks](https://mlsanity.com/paper/2109.10253/) (2021)
- [JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities](https://mlsanity.com/paper/2210.14312/) (2022)
- [Neuro-symbolic partial differential equation solver](https://mlsanity.com/paper/2210.14907/) (2022)
- [chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data](https://mlsanity.com/paper/2501.18715/) (2025)
- [Bringing PDEs to JAX with forward and reverse modes automatic differentiation](https://mlsanity.com/paper/2309.07137/) (2023)

## Abstract (arXiv)

In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a prior for the coefficient to be estimated from noisy data. Neural networks are global, smooth function approximators and as such they do not require explicit regularization of the error functional to recover smooth solutions and coefficients. We give detailed examples using the Poisson equation in 1, 2, and 3 space dimensions and show that the neural network augmentation is robust with respect to noisy and incomplete data, mesh, and geometry.