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This research introduces Staged Depth Training (SDT), a novel curriculum learning method for Physics-Informed Neural Networks (PINNs) that explicitly refines latent representations before increasing network depth. By decoupling representation learning from the final predictor, SDT achieves significant error reductions across diverse physics problems without altering the deployed architecture or inference cost. The findings suggest that explicit control over representation geometry offers a robust pathway to improving PINN performance beyond standard training heuristics.
Staged Depth Training: A Representation Curriculum for PINNs
arXiv:2609.30299v1 Announce Type: new Abstract: Representation quality is a central determinant of PINNs' performance, yet standard training leaves representations to emerge implicitly while fitting the final solution. We introduce \textbf{representation curriculum}, an ordered process in which representations are explicitly learned, transferred independently of their predictors, and progressively refined. We realize it with Staged Depth Training (SDT), which trains a shallow prefix under a temporary physics-informed head, discards the head, and freezes the learned prefix while adding depth, without equation-specific encodings or changes to the final architecture. Across the 20 default forward problems in PINNacle with three backbones, SDT improves 40 of 59 equal-budget problem--backbone cells by at least 5\% and remains within that band in the rest, with a 32.8\% geometric-mean error reduction on a PirateNet-style backbone. Mechanistic ablations suggest that the gain is not explained by optimizer restarts or shallow warm-starting alone. Representation visualizations and hyperparameter-basin analyses provide diagnostic evidence on representation geometry and local sensitivity to shared hyperparameters. On Poisson--Boltzmann 2D, SDT also more than doubles the fitted depth-scaling exponent for both backbones. These results support representation curriculum as a promising training strategy for improving PINNs while preserving the deployed architecture and inference cost.
Abstract:Representation quality is a central determinant of PINNs' performance, yet standard training leaves representations to emerge implicitly while fitting the final solution. We introduce \textbf{representation curriculum}, an ordered process in which representations are explicitly learned, transferred independently of their predictors, and progressively refined. We realize it with Staged Depth Training (SDT), which trains a shallow prefix under a temporary physics-informed head, discards the head, and freezes the learned prefix while adding depth, without equation-specific encodings or changes to the final architecture. Across the 20 default forward problems in PINNacle with three backbones, SDT improves 40 of 59 equal-budget problem--backbone cells by at least 5\% and remains within that band in the rest, with a 32.8\% geometric-mean error reduction on a PirateNet-style backbone. Mechanistic ablations suggest that the gain is not explained by optimizer restarts or shallow warm-starting alone. Representation visualizations and hyperparameter-basin analyses provide diagnostic evidence on representation geometry and local sensitivity to shared hyperparameters. On Poisson--Boltzmann 2D, SDT also more than doubles the fitted depth-scaling exponent for both backbones. These results support representation curriculum as a promising training strategy for improving PINNs while preserving the deployed architecture and inference cost.
Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
arXiv:2609.30299 [cs.LG]
(or arXiv:2609.30299v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.30299 Focus to learn more arXiv-issued DOI via DataCite
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