Loss-Conditional PINNs for Parametric PDE Families
2026-06-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors propose LC-PINN, a new approach where instead of fixing the importance of different physics rules during training, the network learns solutions for many settings at once by treating those importance weights or physical parameters as inputs. This allows the model to handle a whole range of related problems with one training process, instead of retraining for each case. They tested LC-PINN on several physics equations and found it works as well or better than traditional methods that train separately for each setting. Their work combines ideas from physics-informed neural networks and operator learning in a novel way.
Physics-informed neural networks (PINNs)Ordinary differential equations (ODEs)Partial differential equations (PDEs)Loss weightingParametric PDEsOperator learningL-BFGS optimizationHelmholtz equationSchrodinger equationBurgers equation
Authors
Anna Lazareva, Alexander Tarakanov
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.