# Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices

arXiv: https://arxiv.org/abs/2507.09782 · HTML: https://mlsanity.com/paper/2507.09782/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 13 Jul 2025.

## Builds on (most cited first, 11 of 11)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [A Stochastic Quasi-Newton Method for Large-Scale Optimization](https://mlsanity.com/paper/1401.7020/) (2014)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [Stochastic Newton and Cubic Newton Methods with Simple Local Linear-Quadratic Rates](https://mlsanity.com/paper/1912.01597/) (2019)
- [Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states](https://mlsanity.com/paper/2008.00118/) (2020)
- [Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations](https://mlsanity.com/paper/2407.19707/) (2024)
- [A finite difference method with symmetry properties for the high-dimensional Bratu equation](https://mlsanity.com/paper/2410.12553/) (2024)
- [A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations](https://mlsanity.com/paper/2311.11558/) (2023)

## Built on by (most cited first, 3 of 3)

- [Learning with Mandelbrot and Julia](https://mlsanity.com/paper/2509.00903/) (2025)
- [Supervised Metric Regularization Through Alternating Optimization for Multi-Regime Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.09980/) (2026)
- [Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach](https://mlsanity.com/paper/2603.21568/) (2026)

## Abstract (arXiv)

This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg-Marquardt algorithm to optimize network weights for greater accuracy. To enhance computational efficiency in high-dimensional settings, we integrate a stochastic sampling strategy. We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams, incorporating an auxiliary equation to effectively track successive solution branches. For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm-Liouville theory. Numerical experiments are conducted on the discrete Allen-Cahn equation with cubic and quintic nonlinearities in one to five spatial dimensions. The results demonstrate that the proposed approach achieves accuracy comparable to, or better than, traditional numerical methods, especially in high-dimensional regimes where computational resources are a limiting factor. These findings highlight the potential of neural networks as scalable and efficient tools for the study of complex nonlinear lattice systems.