Title: A Curvature-Aware Dynamic Precision Strategy for Physics-Informed Neural Networks

Abstract:

Physics-informed neural networks (PINNs) have emerged as a robust framework for solving partial differential equations (PDEs) by integrating physical laws directly into the training process. However, recent research highlights that the optimization of PINNs is highly sensitive to numerical precision. Standard implementations typically rely on either single precision (FP32), which offers computational efficiency but is susceptible to failure modes, or double precision (FP64), which provides robustness at a significant computational cost. This dichotomy presents a challenging trade-off between processing speed and numerical accuracy.

To address this, we introduce a curvature-aware precision controller that dynamically adjusts numerical precision throughout the training phase, moving away from static implementation choices. By leveraging curvature information extracted from the limited-memory BFGS (L-BFGS) optimizer, our method constructs an adaptive controller. This system maintains FP32 operations when lower precision suffices, but upgrades computations to FP64 when training dynamics reveal numerical sensitivity or precision-induced stagnation.

We validated this approach using four standard benchmarks known for PINN failure modes, alongside an irradiance-driven ordinary differential equation example. Additionally, the method was tested across various neural network architectures. The results demonstrate that the proposed technique consistently achieves accuracy equal to or slightly better than full FP64 training, while significantly reducing training times compared to standard double-precision methods across all tested equations. These findings suggest that precision sensitivity in PINN optimization is dependent on the training phase, and that selectively applying higher precision only during numerically critical stages can reduce computational costs without compromising predictive accuracy.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC