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[2602.21988] Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks
Summary
This article presents a novel method for solving stiff dark matter equations using Jacobian Normalization within Physics-Informed Neural Networks (PINNs), demonstrating improved accuracy over traditional methods.
Why It Matters
Stiff differential equations are a significant challenge in computational physics, particularly in modeling dark matter. This research offers a promising approach that enhances the performance of PINNs, potentially leading to better understanding and predictions in high-energy physics and cosmology.
Key Takeaways
- Jacobian Normalization improves convergence in PINNs for stiff equations.
- The method shows higher accuracy than existing attention mechanisms.
- Successful application to Boltzmann equations governing dark matter.
- Validates performance through an inverse problem using observed relic density.
- Offers a hyperparameter-free solution to a complex computational challenge.
High Energy Physics - Phenomenology arXiv:2602.21988 (hep-ph) [Submitted on 25 Feb 2026] Title:Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks Authors:M. P. Bento, H. B. Câmara, J. R. Rocha, J. F. Seabra View a PDF of the paper titled Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks, by M. P. Bento and 3 other authors View PDF HTML (experimental) Abstract: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. Comments: Subjects: High Energy Physics - Phenomenology (hep-ph...