New Quasi-Newton Method Improves PINN Training Accuracy

Training Physics-informed neural networks (PINNs) to solve partial differential equations often requires highly accurate quasi-Newton methods, but these models frequently encounter challenges due to the complex, often indefinite or poorly scaled curvature of their objective functions. A new method, SCORE (Self-Concordance-inspired Quasi-Newton Regularization), has been developed to overcome these limitations. SCORE combines principles from regularized quasi-Newton methods and self-concordant methods to stabilize secant models and guide step selection. The core innovation of SCORE lies in its decrement-coupled shifted secant geometry. A single quasi-Newton decrement simultaneously determines a candidate step and an adaptive shift for defining the next secant geometry, without needing to construct or compute Hessian-vector products. Experiments across various complex equations, including Burgers, Kuramoto-Sivashinsky, and Ginzburg-Landau, demonstrate that SCORE consistently achieves lower final errors compared to standard BFGS and self-scaled Broyden baselines, indicating a significant improvement in high-accuracy refinement for PINNs.