Double screening in the training dynamics of variational physics-informed neural networks for heterogeneous coupled parabolic systems

Abstract

We analyze the training dynamics of variational physics-informed neural networks applied to linear coupled parabolic convection--diffusion--reaction systems, called heterogeneous when only a subset of the components undergoes convective transport. In the neural tangent kernel regime, the gradient flow on the space-time variational residuals reduces to a linear differential system whose operator is a Gram matrix built from the space-time symbol of the system operator and the matrix tangent kernel. The main result is a double screening theorem. Under dominant convection, the Schur complement of this matrix relative to the block of convective components converges to an expression that involves only the diffusive block of the symbol, with no coupling term, together with the Schur complement of the tangent kernel. From this we derive four consequences, namely an exact identity quantifying the screened coupling energy, a degradation law for the training rate governed by the canonical correlations of the kernel, a bound on the condition number in terms of the P\'eclet number, and the non-participation of temporal frequencies in the screening mechanism. The growth of the condition number slows down shared-step gradient descent, whereas the continuous flow suffers no slowdown, which makes the training difficulty attributable to the optimizer rather than to the approximation. We finally show that the Adam optimizer, through its adaptive scaling, mitigates this difficulty provided that the architecture separates the parameters associated with the convective and diffusive components, confirming the architectural prescription that follows from the second screening. The predictions are validated numerically down to machine precision, on a two-dimensional exchanger, and by the full training of finite-width networks.