MULTI-DIMENSIONAL TRAINING-PRIORITY WEIGHTING BASED ON PHYSICAL INFORMATION PROPAGATION PATHS: A UNIFIED RESIDUAL-WEIGHTING FRAMEWORK FOR PHYSICS-INFORMED NEURAL NETWORKS

The Problem with Synchronous Optimization

Physics-Informed Neural Networks (PINNs) typically minimize a loss function that aggregates residuals from all spatial and temporal domains simultaneously. This approach assumes that all physical constraints can be learned in parallel. However, many physical systems—such as those governed by convection, wave propagation, or boundary-driven diffusion—evolve along a specific 'source-to-response' path. The authors argue that standard PINNs fail to respect this natural order, leading to training instability and poor accuracy because they treat premise regions (the 'source') and dependent regions (the 'response') with equal priority.

Theoretical Analysis via NTK

Using Neural Tangent Kernel (NTK) dynamics, the researchers demonstrate that the convergence order of a standard PINN is governed by the NTK spectrum, which is determined by the network architecture and collocation points rather than the underlying physics. Because the NTK spectrum does not inherently favor premise regions, the network does not naturally learn the solution in the order dictated by the physical propagation path. This spectral bias explains why standard PINNs often struggle with problems where the solution must form progressively.

A Unified Priority Framework

To rectify this, the authors propose a multi-dimensional weighting framework that explicitly incorporates the physical propagation path into the loss function. By partitioning the domain along the propagation path and applying negative-exponential weights to the residuals, the method forces the network to satisfy premise regions before moving on to dependent regions. For problems with multiple, potentially conflicting propagation directions, the authors introduce a 'directional compatibility coefficient' to determine how these priorities should be coupled—specifically, that orthogonal directions can be coupled multiplicatively, while coaxial opposite directions cannot.

This framework provides a systematic way to improve PINN performance without modifying the underlying network architecture. By aligning the training process with the physical reality of the PDE, the method enhances convergence behavior and prediction accuracy in problems with clear propagation paths, such as transport or boundary-dominated steady-state equations, while maintaining controllable computational costs.