Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion

• URL: http://arxiv.org/abs/2606.07982v1
• Date: Sat, 06 Jun 2026 05:13:47 GMT
• Title: Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion
• Abstract summary: High-dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods.<n>This paper presents a Physics-Informed Neural Network (PINN) framework as a systematic solution to this problem.<n>PINN sustains approximately 91% accuracy while Finite Difference Method (FDM) collapses to 36%, a clear decisive advantage.
• Score: 1.4740287473718565
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: High-dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods: accuracy degrades catastrophically where physical noise is unavoidable. This paper presents a Physics-Informed Neural Network (PINN) framework as a systematic solution to this problem across one, two, and three spatial dimensions, establishing clear operational regimes that redefine solver selection in noisy thermal systems. Under 20% boundary noise in 3D, PINN sustains approximately 91% accuracy while Finite Difference Method (FDM) collapses to 36%, a clear decisive advantage. This is further confirmed in a physical copper thermal system, where PINN reduces boundary reconstruction error by 3.3 times under realistic noise conditions. This noise resilience is accompanied by a dimensionality-driven efficiency crossover: PINN requires fewer spacetime nodes than FDM in 3D while achieving superior accuracy, exposing the true cost of classical discretization at scale. These findings reframe solver selection: the decisive axis is not accuracy alone, but noise exposure and dimensionality jointly. When noise and dimensionality are both high, the classical solver paradigm is insufficient; this work provides the foundation to justify PINN as the operational standard in such regimes.

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