Physics-constrained neural networks for direct parameter identification under model-form uncertainty

Abstract

Parameter identification in nonlinear dynamical systems is complicated by model-form uncertainty arising from systematic biases that violate the zero-mean error assumption of standard data assimilation methods. Recent neural-network-based approaches learn arbitrary bias corrections online. However, they require careful regularization to ensure unique solutions and carry computational overhead from ensemble propagation and in-situ training. We present a framework that integrates the parametrized-background data-weak formulation with attention-based parameter identification networks. It projects model error onto a dictionary of physically motivated spatial templates rather than learning arbitrary corrections. This provides uniqueness through hard subspace constraints rather than soft regularization penalties, albeit at the cost of restricting the representable bias space to patterns anticipated from domain knowledge. The framework is demonstrated on parameter estimation in the Rijke tube model, achieving robust generalization to out-of-distribution (OOD) bias patterns not seen during training. Ablation studies confirm the importance of state supervision for amplitude-sensitive parameters and temporal bias modeling for phase-sensitive parameters. Performance degrades gradually when bias patterns lie outside the template span. This indicates that the attention encoder extracts parameter information from bias-invariant features rather than relying critically on template-based bias capture. The physics-constrained and universal approximator approaches represent complementary points on the flexibility–efficiency tradeoff, suited to different operational contexts.