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

arXiv: https://arxiv.org/abs/2606.07982 · HTML: https://mlsanity.com/paper/2606.07982/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 32 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 16 of 16)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks](https://mlsanity.com/paper/2110.13330/) (2021)
- [Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equations](https://mlsanity.com/paper/2407.10836/) (2024)
- [Robustness of Physics-Informed Neural Networks to Noise in Sensor Data](https://mlsanity.com/paper/2211.12042/) (2022)
- [Noisy PDE Training Requires Bigger PINNs](https://mlsanity.com/paper/2507.06967/) (2025)
- [Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems](https://mlsanity.com/paper/2509.20191/) (2025)
- [A globalization of L-BFGS and the Barzilai-Borwein method for nonconvex unconstrained optimization](https://mlsanity.com/paper/2401.03805/) (2024)
- [Robust Physics Discovery from Highly Corrupted Data: A PINN Framework Applied to the Nonlinear Schr"odinger Equation](https://mlsanity.com/paper/2601.04176/) (2026)
- [naPINN: Noise-Adaptive Physics-Informed Neural Networks for Recovering Physics from Corrupted Measurement](https://mlsanity.com/paper/2602.02547/) (2026)

## Abstract (arXiv)

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.