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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.12556 |
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| _version_ | 1866912964529881088 |
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| author | Chaudhry, Faris |
| author_facet | Chaudhry, Faris |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12556 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scaling Laws and Pathologies of Single-Layer PINNs: Network Width and PDE Nonlinearity Chaudhry, Faris Machine Learning Numerical Analysis Computational Physics 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. |
| title | Scaling Laws and Pathologies of Single-Layer PINNs: Network Width and PDE Nonlinearity |
| topic | Machine Learning Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2603.12556 |