Component-Level Evaluation of Adaptive PINN Training for CFD-Oriented Crystal Growth Simulation

For systems with steep gradients, sharp interfaces, or severe spatio-temporal coupling, Physics-informed neural networks (PINNs) suffer from spectral bias, geometric inflexibility, and boundary constraint conflicts, which undermine accuracy and convergence. To overcome these issues, we propose a geometry-adaptive and constraint-enhanced PINN (GAC-PINN). The framework comprises four components: a gradient-driven adaptive grid mapping (AGM) for diffeomorphic point concentration with Jacobian regularization, an adaptive bandwidth hard-constraint ansatz with spatially-varying boundary transition widths, a Gaussian Fourier feature mapping as a spectral preconditioner to further enhance high-wavenumber representation, and an operator-aware router that automatically selects the appropriate hard-constraint construction based on whether the governing PDE contains temporal derivatives. An AGM callback mechanism and a three-stage training strategy ensure stable coordination. Benchmarks including the viscous Burgers equation, a sharp-peaked 2D Poisson problem, and the Allen-Cahn phase-transition equation show that GAC-PINN attains relative (L^2) errors of ((1.747\pm 0.450)\times 10^{-4}), ((2.868\pm 0.947)\times 10^{-5}), and ((1.756 \pm 0.712)\times 10^{-3}), respectively, consistently outperforming the baselines. Ablation studies further reveal that AGM alone yields a substantially lower error than residual-based adaptive refinement (RAR), while RAR becomes beneficial only when combined with FFM, demonstrating a context-dependent module interaction. Convergence analysis verifies rapid error reduction and saturation with increasing resolution, establishing a practical adaptive framework for high-fidelity simulation of problems with localized sharp features in applied mechanics and computational physics.

Physics-informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems governed by partial differential equations (PDEs), but standard PINN training typically relies on soft penalty formulations that combine PDE residuals, data mismatch, and initial/boundary conditions using manually chosen weights. This often leads to ill-conditioning, sensitivity to loss weights, and poor constraint satisfaction. In this work, we reformulate PINN training as an equality-constrained optimization problem and propose a novel Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN (AdamFLIP). The key idea is to view the constraint residuals as the output of a controlled dynamical system and to compute the Lagrange multiplier as a feedback input that locally drives these residuals toward stable linear contraction dynamics. AdamFLIP then applies Adam-style first- and second-moment adaptation to the resulting feedback-linearized Lagrangian gradient, combining principled constraint handling with the scalability and robustness of adaptive neural-network optimization. We test AdamFLIP on a range of benchmark forward and inverse PDE problem, and it consistently outperforms both the standard soft-constrained PINN and state-of-the-art constrained optimizers. Specifically, on the Navier--Stokes equations benchmark, AdamFLIP \textbf{reduces relative $L_2$ error by more than two thirds} for the predicted solution compared to the next best method. Our AdamFLIP framework provides an effective and computationally scalable hard constraint optimization method for PINN training.

Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.