Stavros Kassinos | Searcharxiv

arXiv2026

Auxiliary Finite-Difference Residual-Gradient Regularization for PINNs

Physics-informed neural networks (PINNs) are often selected by a single scalar loss even when the quantity of interest is more specific. We study a hybrid design in which the governing PDE residual remains automatic-differentiation (AD) based, while finite differences (FD) appear only in a weak auxiliary term that penalizes gradients of the sampled residual field. The FD term regularizes the residual field without replacing the PDE residual itself. We examine this idea in two stages. Stage 1 is a controlled Poisson benchmark comparing a baseline PINN, the FD residual-gradient regularizer, and a matched AD residual-gradient baseline. Stage 2 transfers the same logic to a three-dimensional annular heat-conduction benchmark (PINN3D), where baseline errors concentrate near a wavy outer wall and the auxiliary grid is implemented as a body-fitted shell adjacent to the wall. In Stage 1, the FD regularizer reproduces the main effect of residual-gradient control while exposing a trade-off between field accuracy and residual cleanliness. In Stage 2, the shell regularizer improves the application-facing quantities, namely outer-wall flux and boundary-condition behavior. Across seeds 0-5 and 100k epochs, the most reliable tested configuration is a fixed shell weight of 5e-4 under the Kourkoutas-beta optimizer regime: relative to a matched run without the shell term, it reduces the mean outer-wall BC RMSE from 1.22e-2 to 9.29e-4 and the mean wall-flux RMSE from 9.21e-3 to 9.63e-4. Adam with beta2=0.999 becomes usable when the initial learning rate is reduced to 1e-3, although its shell benefit is less robust than under Kourkoutas-beta. Overall, the results support a targeted view of hybrid PINNs: an auxiliary-only FD regularizer is most valuable when it is aligned with the physical quantity of interest, here the outer-wall flux.

cs.LG↗

arXiv2024

Beyond Language: Applying MLX Transformers to Engineering Physics

Transformer Neural Networks are driving an explosion of activity and discovery in the field of Large Language Models (LLMs). In contrast, there have been only a few attempts to apply Transformers in engineering physics. Aiming to offer an easy entry point to physics-centric Transformers, we introduce a physics-informed Transformer model for solving the heat conduction problem in a 2D plate with Dirichlet boundary conditions. The model is implemented in the machine learning framework MLX and leverages the unified memory of Apple M-series processors. The use of MLX means that the models can be trained and perform predictions efficiently on personal machines with only modest memory requirements. To train, validate and test the Transformer model we solve the 2D heat conduction problem using central finite differences. Each finite difference solution in these sets is initialized with four random Dirichlet boundary conditions, a uniform but random internal temperature distribution and a randomly selected thermal diffusivity. Validation is performed in-line during training to monitor against over-fitting. The excellent performance of the trained model is demonstrated by predicting the evolution of the temperature field to steady state for the unseen test set of conditions.

cs.CE↗