Curvature-aware dynamic precision approach for physics-informed neural networks

2026-06-03

AI Summary

This work addresses the longstanding trade-off in physics-informed neural network (PINN) training between the computational efficiency of single-precision (FP32) arithmetic and the numerical stability of double-precision (FP64). The authors propose a curvature-aware dynamic precision control method that, for the first time, leverages curvature information from the L-BFGS optimizer to adaptively switch between FP32 and FP64 during training, enabling on-demand high-precision computation. By moving beyond fixed-precision training, this approach achieves accuracy comparable to or slightly better than full FP64 across multiple canonical PINN failure cases and ordinary differential equation (ODE) benchmarks, while substantially reducing overall training time.

Abstract

Physics-informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, recent studies show that PINN optimisation is sensitive to numerical precision. Existing implementations commonly use either single precision (FP32), which is computationally efficient but prone to failure modes, or double precision (FP64), which is robust but substantially expensive. This creates a trade-off between computational efficiency and numerical accuracy. To reduce the computational cost of double-precision training while retaining prediction accuracy, we propose a curvature-aware precision controller that adapts numerical precision during training rather than treating it as a fixed implementation choice. The proposed method reuses curvature information derived from the limited-memory BFGS (L-BFGS) optimiser to construct a precision controller, retaining FP32 when lower precision is sufficient and promoting computation to FP64 when the training dynamics indicate numerical sensitivity or precision-limited stagnation. We evaluate the proposed approach on four canonical PINN failure-mode benchmarks and an irradiance-driven ordinary differential equation example. We further test the proposed approach across different neural network architectures. The method consistently matches or even slightly exceeds full FP64 solution accuracy while reducing training time relative to full double-precision training on all benchmark equations. The obtained results indicate that precision sensitivity in PINN optimisation is phase-dependent, and that selectively applying higher precision only during numerically critical stages can lower computational cost without sacrificing predictive accuracy.

Problem

• physics-informed neural networks
• numerical precision
• computational efficiency
• optimization
• partial differential equations

Innovation

• curvature-aware
• dynamic precision
• physics-informed neural networks
• numerical accuracy
• adaptive computation