Loss-Conditional PINNs for Parametric PDE Families

• URL: http://arxiv.org/abs/2606.04420v1
• Date: Wed, 03 Jun 2026 04:02:10 GMT
• Title: Loss-Conditional PINNs for Parametric PDE Families
• Abstract summary: Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses.<n>We introduce LC-PINN, which adapts the lossconditional training of Dosovitskiy and Djolonga ( 2020) to the PDE-residual setting.<n>It stays fully physics-informed but amortises training over a parametric family.
• Score: 45.945826972787415
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses. Their performance is often dominated by the choice of loss weights: a poor weighting can drive training to a degenerate solution in which one physical constraint is satisfied while another is ignored. Existing methods select or adapt a single good set of weights. We take a different view: instead of tuning one weight vector, we explore the entire weight space during training. We introduce LC-PINN, which adapts the loss-conditional training of Dosovitskiy and Djolonga (2020) to the PDE-residual setting: the conditioning vector (either the loss weights or a scalar physical coefficient) is treated as a network input and sampled from a simple prior at every optimisation step. This turns PINN training into learning a continuous family of solutions indexed by that vector, with no solver-generated paired data. LC-PINN thus lies between classical PINNs and operator learning: it stays fully physics-informed but amortises training over a parametric family. Our contribution is not the loss-conditional construction itself, but its extension to PINNs, the unification of the loss-weight and parametric-coefficient regimes under one architecture (concatenation for loss weights, FiLM for coefficients), and a fixed-quadrature L-BFGS finishing protocol that makes the parametric-coefficient regime trainable. We give a lambda-invariance result for the conditional optimum and study LC-PINN on parametric Helmholtz, Schrodinger, viscous Burgers, and Buckley-Leverett equations. A single LC-PINN matches or improves retrained per-weight PINN baselines while parameterising the full family in one model, at a total cost that amortises favourably against per-instance retraining.

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