Partial-Moment PINNs for Caldeira--Leggett Parameter Learning in Quantum Brownian Motion

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.

Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a family of PDE solutions over a parameter domain. Existing ParamPINNs, however, still face inefficient training, uneven accuracy across parameters, and overfitting to a limited set of sampled parameter tasks, which can impair generalization to unsampled parameters. To address these issues, we propose a continual-learning physics-informed neural network (CL-PINN), which treats PDE instances at different parameter values as related tasks and learns them sequentially. CL-PINN combines Bayesian-optimization-based active parameter selection, task-wise dynamic loss weighting, sparse physics-constrained replay, and an optional parameter subnetwork to improve task allocation and knowledge retention under bounded active-task capacity. It requires no observational data and is designed to solve parameterized PDEs over relatively broad parameter domains under limited computational resources. Multi-seed evaluations on five benchmarks, including one continuous function and four parameterized PDEs, show that Bayesian selection substantially reduces objective-loss queries relative to grid-greedy search, while sparse replay mitigates forgetting of earlier tasks. Under the prescribed within-case resource protocols, CL-PINN generally provides higher and more balanced solution accuracy than fixed-sampling and grid-greedy baselines. CL-PINN offers a practical route toward learning PDE solutions that generalize across physical parameters and has the potential to support reusable physics-informed surrogates for large-scale engineering parameter studies.

Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to $23\times$ fewer collocation points and a vanilla architecture.