Title: Loss-Conditional PINNs for Parametric PDE Families

Abstract:

Physics-informed neural networks (PINNs) estimate solutions to ordinary and partial differential equations (ODEs and PDEs) by minimizing a composite loss function that typically combines residuals, boundary conditions, initial states, and data fidelity. However, the efficacy of these models frequently hinges on the selection of loss weights; suboptimal weighting can lead to degenerate training outcomes where specific physical constraints are fulfilled while others are neglected. While conventional approaches focus on identifying or adapting a single optimal set of weights, we propose an alternative strategy: exploring the full weight space throughout the training process.

We present LC-PINN, a method that adapts loss-conditional training—originally introduced by Dosovitskiy and Djolonga (2020)—to the context of PDE residuals. In this framework, the conditioning vector, which may consist of loss weights or a scalar physical coefficient, is incorporated as a network input and sampled from a simple prior at each optimization step. This approach transforms PINN training into the task of learning a continuous family of solutions indexed by the conditioning vector, eliminating the need for solver-generated paired data. Consequently, LC-PINN occupies a middle ground between traditional PINNs and operator learning: it retains full physics-informed characteristics while amortizing training costs across a parametric family.

Our primary contribution lies not in the loss-conditional structure itself, but in its application to PINNs. We unify the loss-weight and parametric-coefficient regimes within a single architecture, employing concatenation for loss weights and FiLM for coefficients. Additionally, we introduce a fixed-quadrature L-BFGS finishing protocol to ensure the trainability of the parametric-coefficient regime. We also provide a lambda-invariance result for the conditional optimum. Empirical evaluations on parametric Helmholtz, Schrödinger, viscous Burgers, and Buckley-Leverett equations demonstrate that a single LC-PINN model matches or outperforms retrained per-weight PINN baselines. By parameterizing the entire family within one model, LC-PINN achieves a total computational cost that amortizes favorably compared to retraining individual instances.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC