# Scaling Laws and Pathologies of Single-Layer PINNs: Network Width and PDE Nonlinearity

arXiv: https://arxiv.org/abs/2603.12556 · HTML: https://mlsanity.com/paper/2603.12556/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 16 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 8 of 8)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.03864/) (2024)
- [The Effect of Network Width on the Performance of Large-batch Training](https://mlsanity.com/paper/1806.03791/) (2018)

## Built on by (most cited first, 1 of 1)

- [Variational Boosting for Physics-Informed Neural Networks](https://mlsanity.com/paper/2607.23940/) (2026)

## Abstract (arXiv)

We establish empirical scaling laws for Single-Layer Physics-Informed Neural Networks on canonical nonlinear PDEs. We identify a dual optimization failure: (i) a baseline pathology, where the solution error fails to decrease with network width, even at fixed nonlinearity, falling short of theoretical approximation bounds, and (ii) a compounding pathology, where this failure is exacerbated by nonlinearity. We provide quantitative evidence that a simple separable power law is insufficient, and that the scaling behavior is governed by a more complex, non-separable relationship. This failure is consistent with the concept of spectral bias, where networks struggle to learn the high-frequency solution components that intensify with nonlinearity. We show that optimization, not approximation capacity, is the primary bottleneck, and propose a methodology to empirically measure these complex scaling effects.