# Large-scale Neural Solvers for Partial Differential Equations

arXiv: https://arxiv.org/abs/2009.03730 · HTML: https://mlsanity.com/paper/2009.03730/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Sep 2020.

## Built on by (most cited first, 17 of 17)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies](https://mlsanity.com/paper/2306.17648/) (2023)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions](https://mlsanity.com/paper/2509.05117/) (2025)
- [Learning Runge-Kutta Integration Schemes for ODE Simulation and Identification](https://mlsanity.com/paper/2105.04999/) (2021)
- [Data-driven Science and Machine Learning Methods in Laser-Plasma Physics](https://mlsanity.com/paper/2212.00026/) (2022)
- [Adapting Physics-Informed Neural Networks to Improve ODE Optimization in Mosquito Population Dynamics](https://mlsanity.com/paper/2406.05108/) (2024)
- [AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition](https://mlsanity.com/paper/2510.08924/) (2025)
- [Invertible Surrogate Models: Joint surrogate modelling and reconstruction of Laser-Wakefield Acceleration by invertible neural networks](https://mlsanity.com/paper/2106.00432/) (2021)
- [Continual learning autoencoder training for a particle-in-cell simulation via streaming](https://mlsanity.com/paper/2211.04770/) (2022)
- [PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks](https://mlsanity.com/paper/2504.19013/) (2025)
- [Hybrid machine learning based scale bridging framework for permeability prediction of fibrous structures](https://mlsanity.com/paper/2502.05044/) (2025)
- [Learning Electron Bunch Distribution along a FEL Beamline by Normalising Flows](https://mlsanity.com/paper/2303.00657/) (2023)
- [When Does Frequency Decomposition Benefit Physics-Informed Neural Networks? A Preliminary Ablation Study](https://mlsanity.com/paper/2608.24940/) (2026)

## Abstract (arXiv)

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. Scanning the parameters of the underlying model significantly increases the runtime as the simulations have to be cold-started for each parameter configuration. Machine Learning based surrogate models denote promising ways for learning complex relationship among input, parameter and solution. However, recent generative neural networks require lots of training data, i.e. full simulation runs making them costly. In contrast, we examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) solely requiring initial/boundary values and validation points for training but no simulation data. The induced curse of dimensionality is approached by learning a domain decomposition that steers the number of neurons per unit volume and significantly improves runtime. Distributed training on large-scale cluster systems also promises great utilization of large quantities of GPUs which we assess by a comprehensive evaluation study. Finally, we discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.