# Mesh-free Eulerian Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2206.01545 · HTML: https://mlsanity.com/paper/2206.01545/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 51 references that are in the map.
- First posted on arXiv 3 Jun 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [tfp.mcmc: Modern Markov Chain Monte Carlo Tools Built for Modern Hardware](https://mlsanity.com/paper/2002.01184/) (2020)
- [Quasi-Monte Carlo sampling for machine-learning partial differential equations](https://mlsanity.com/paper/1911.01612/) (2019)
- [Optimal Transport Based Refinement of Physics-Informed Neural Networks](https://mlsanity.com/paper/2105.12307/) (2021)

## Built on by (most cited first, 4 of 4)

- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Lagrangian Flow Networks for Conservation Laws](https://mlsanity.com/paper/2305.16846/) (2023)
- [Advancing Generalization in PINNs through Latent-Space Representations](https://mlsanity.com/paper/2411.19125/) (2024)
- [Physics-guided Full Waveform Inversion using Encoder-Solver Convolutional Neural Networks](https://mlsanity.com/paper/2405.17696/) (2024)

## Abstract (arXiv)

Physics-informed Neural Networks (PINNs) have recently emerged as a principled way to include prior physical knowledge in form of partial differential equations (PDEs) into neural networks. Although PINNs are generally viewed as mesh-free, current approaches still rely on collocation points within a bounded region, even in settings with spatially sparse signals. Furthermore, if the boundaries are not known, the selection of such a region is difficult and often results in a large proportion of collocation points being selected in areas of low relevance. To resolve this severe drawback of current methods, we present a mesh-free and adaptive approach termed particle-density PINN (pdPINN), which is inspired by the microscopic viewpoint of fluid dynamics. The method is based on the Eulerian formulation and, different from classical mesh-free method, does not require the introduction of Lagrangian updates. We propose to sample directly from the distribution over the particle positions, eliminating the need to introduce boundaries while adaptively focusing on the most relevant regions. This is achieved by interpreting a non-negative physical quantity (such as the density or temperature) as an unnormalized probability distribution from which we sample with dynamic Monte Carlo methods. The proposed method leads to higher sample efficiency and improved performance of PINNs. These advantages are demonstrated on various experiments based on the continuity equations, Fokker-Planck equations, and the heat equation.