From Numerical Schemes to Neural Network Architectures: A Physics-Informed Framework for Differential Equations

Abstract

Physics-informed neural networks usually incorporate differential equations primarily through the loss function, while the neural architecture itself is selected independently of the underlying numerical structure. This work develops a scheme-induced approach in which an analytically modified numerical method defines the computational architecture, while selected elements of the numerical scheme are made trainable and refined using physics-informed loss functions. The specific contribution is the formulation and systematic investigation of a class of scheme-parameterized low-fidelity physics-informed neural networks (LF-PINNs) for evolutionary PDEs. In the implementation studied here, a compact parameter network directly controls the coefficient of a generalized θ-scheme, while the implicit update is represented by a fixed number of Picard iterations. The approach is evaluated on six one-dimensional PDE benchmark configurations against conventional, hard-constrained, and adaptively weighted PINNs. The nine-parameter LF-PINN achieves competitive accuracy on several benchmark problems, while its trainable parameter remains explicitly connected to a coefficient of the underlying numerical scheme. The results also show that performance is problem-dependent and that a small number of trainable parameters does not necessarily imply lower computational cost. Stability, ablation, computational-cost, and generalization analyses further characterize the properties and limitations of the scheme-induced construction.