<p>We present an Iterative Refinement Physics-Informed Neural Network (IR-PINN) framework that enables scientific machine learning on low-precision AI hardware. This approach overcomes the fundamental precision limitations of FP16 arithmetic for partial differential equation (PDE) solving. IR-PINN decomposes the solution into a compact low-precision base network and a sequence of lightweight correction networks that learn progressively smaller-magnitude residuals. While corrections are computed in FP16 for hardware efficiency, accumulation and precomputed PDE derivatives are maintained in full precision, creating a numerically stable hierarchy robust to rounding errors. Comprehensive evaluation across four canonical PDE benchmarks (2D Poisson, 2D Helmholtz, 1D Burgers’, and stiff 1D advection–diffusion) demonstrates that IR-PINN restores FP32-level accuracy using only FP16 computations. Results averaged over 20 independent runs show FP16 IR-PINN reduces RMSE by factors of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(10.7\times \)</EquationSource> </InlineEquation> (Poisson), <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(4.0\times \)</EquationSource> </InlineEquation> (Helmholtz), and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(2.6\times \)</EquationSource> </InlineEquation> (Burgers) relative to vanilla FP16. Critically, IR-PINN matches or exceeds FP32 baseline performance, achieving up to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(3.9\times \)</EquationSource> </InlineEquation> (Poisson) and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(3.4\times \)</EquationSource> </InlineEquation> (Helmholtz) improvement over FP32 while maintaining Burgers’ accuracy. For the stiff advection–diffusion problem, IR-PINN attains <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(3.5\times \)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(1.4\times \)</EquationSource> </InlineEquation> RMSE reduction versus FP16 and FP32 respectively. Statistical significance (two-sided Student’s <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(t\)</EquationSource> </InlineEquation>-test, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\alpha =0.05\)</EquationSource> </InlineEquation>) is established for Poisson, Helmholtz, and advection–diffusion improvements. IR-PINN thus provides a foundational methodology to bridge the precision gap between modern AI hardware capabilities and scientific computing requirements, enabling high-fidelity PDE solutions on emerging low-precision accelerators.</p>

Accuracy Recovery for Low-Precision PINNs: An Iterative Refinement Framework

期刊论文
发表日期：

2026年1月20日

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Mikhail E. Smorkalov,
Bari R. Khairullin,
Sergey G. Rykovanov

摘要

We present an Iterative Refinement Physics-Informed Neural Network (IR-PINN) framework that enables scientific machine learning on low-precision AI hardware. This approach overcomes the fundamental precision limitations of FP16 arithmetic for partial differential equation (PDE) solving. IR-PINN decomposes the solution into a compact low-precision base network and a sequence of lightweight correction networks that learn progressively smaller-magnitude residuals. While corrections are computed in FP16 for hardware efficiency, accumulation and precomputed PDE derivatives are maintained in full precision, creating a numerically stable hierarchy robust to rounding errors. Comprehensive evaluation across four canonical PDE benchmarks (2D Poisson, 2D Helmholtz, 1D Burgers’, and stiff 1D advection–diffusion) demonstrates that IR-PINN restores FP32-level accuracy using only FP16 computations. Results averaged over 20 independent runs show FP16 IR-PINN reduces RMSE by factors of \(10.7\times \) (Poisson), \(4.0\times \) (Helmholtz), and \(2.6\times \) (Burgers) relative to vanilla FP16. Critically, IR-PINN matches or exceeds FP32 baseline performance, achieving up to \(3.9\times \) (Poisson) and \(3.4\times \) (Helmholtz) improvement over FP32 while maintaining Burgers’ accuracy. For the stiff advection–diffusion problem, IR-PINN attains \(3.5\times \) and \(1.4\times \) RMSE reduction versus FP16 and FP32 respectively. Statistical significance (two-sided Student’s \(t\) -test, \(\alpha =0.05\) ) is established for Poisson, Helmholtz, and advection–diffusion improvements. IR-PINN thus provides a foundational methodology to bridge the precision gap between modern AI hardware capabilities and scientific computing requirements, enabling high-fidelity PDE solutions on emerging low-precision accelerators.