# Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2602.21988 · HTML: https://mlsanity.com/paper/2602.21988/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 25 Feb 2026.

## Builds on (most cited first, 18 of 18)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [A Living Review of Machine Learning for Particle Physics](https://mlsanity.com/paper/2102.02770/) (2021)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks](https://mlsanity.com/paper/2201.06780/) (2022)
- [Critical Investigation of Failure Modes in Physics-informed Neural Networks](https://mlsanity.com/paper/2206.09961/) (2022)
- [VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior](https://mlsanity.com/paper/2406.06287/) (2024)
- [Stiff Transfer Learning for Physics-Informed Neural Networks](https://mlsanity.com/paper/2501.17281/) (2025)
- [Learning solutions of parametric Navier-Stokes with physics-informed neural networks](https://mlsanity.com/paper/2402.03153/) (2024)
- [Accelerating lensed quasar discovery and modeling with physics-informed variational autoencoders](https://mlsanity.com/paper/2412.12709/) (2024)
- [Unraveling particle dark matter with Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.17597/) (2025)

## Abstract (arXiv)

Stiff differential equations pose a major challenge for Physics-Informed Neural Networks (PINNs), often causing poor convergence. We propose a simple, hyperparameter-free method to address stiffness by normalizing loss residuals with the Jacobian. We provide theoretical indications that Jacobian-based normalization can improve gradient descent and validate it on benchmark stiff ordinary differential equations. We then apply it to a realistic system: the stiff Boltzmann equations (BEs) governing weakly interacting massive particle (WIMP) dark matter (DM). Our approach achieves higher accuracy than attention mechanisms previously proposed for handling stiffness, recovering the full solution where prior methods fail. This is further demonstrated in an inverse problem with a single experimental data point - the observed DM relic density - where our inverse PINNs correctly infer the cross section that solves the BEs in both Standard and alternative cosmologies.