# Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity

arXiv: https://arxiv.org/abs/2203.08802 · HTML: https://mlsanity.com/paper/2203.08802/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 15 Mar 2022.

## Builds on (most cited first, 6 of 6)

- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture](https://mlsanity.com/paper/2005.00615/) (2020)
- [PSO-PINN: Physics-Informed Neural Networks Trained with Particle Swarm Optimization](https://mlsanity.com/paper/2202.01943/) (2022)
- [Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem](https://mlsanity.com/paper/2001.05172/) (2020)
- [Physics-informed attention-based neural network for solving non-linear partial differential equations](https://mlsanity.com/paper/2105.07898/) (2021)

## Built on by (most cited first, 22 of 22)

- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread](https://mlsanity.com/paper/2206.12625/) (2022)
- [Binary structured physics-informed neural networks for solving equations with rapidly changing solutions](https://mlsanity.com/paper/2401.12806/) (2024)
- [Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems](https://mlsanity.com/paper/2212.14012/) (2022)
- [A Critical Review of Physics-Informed Machine Learning Applications in Subsurface Energy Systems](https://mlsanity.com/paper/2308.04457/) (2023)
- [Transfer learning-based physics-informed convolutional neural network for simulating flow in porous media with time-varying controls](https://mlsanity.com/paper/2310.06319/) (2023)
- [PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs](https://mlsanity.com/paper/2405.08558/) (2024)
- [GRINN: A Physics-Informed Neural Network for solving hydrodynamic systems in the presence of self-gravity](https://mlsanity.com/paper/2308.08010/) (2023)
- [LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blocks](https://mlsanity.com/paper/2508.08935/) (2025)
- [Asymptotic-Preserving Neural Networks for hyperbolic systems with diffusive scaling](https://mlsanity.com/paper/2210.09081/) (2022)
- [Neural Operator-Based Proxy for Reservoir Simulations Considering Varying Well Settings, Locations, and Permeability Fields](https://mlsanity.com/paper/2407.09728/) (2024)
- [A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I](https://mlsanity.com/paper/2307.04121/) (2023)
- [Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networks](https://mlsanity.com/paper/2307.06362/) (2023)
- [Label Propagation for Physics-Informed Neural Networks and Physics-Informed Gaussian Processes](https://mlsanity.com/paper/2404.05817/) (2024)
- [Fine-Tuning Flow Matching via Maximum Likelihood Estimation of Reconstructions](https://mlsanity.com/paper/2510.02081/) (2025)
- [Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media](https://mlsanity.com/paper/2511.13734/) (2025)
- [A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws](https://mlsanity.com/paper/2604.15617/) (2026)
- [SPLIT-PINN: Separable Probability Learning Technique via Physics-Informed Neural Networks for High-Dimensional Probabilistic Modeling](https://mlsanity.com/paper/2606.04000/) (2026)
- [Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables](https://mlsanity.com/paper/2607.14855/) (2026)
- [Physics Informed Random Feature Neural Networks for Solving PDEs](https://mlsanity.com/paper/2609.16406/) (2026)
- [Adaptive Physics-Informed Neural Networks for the Blasius Boundary-Layer Problem](https://mlsanity.com/paper/2609.22185/) (2026)
- [Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks](https://mlsanity.com/paper/2610.02230/) (2026)

## Abstract (arXiv)

Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain "stiff" problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method.