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FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries

Thivin Anandh, Divij Ghose, Himanshu Jain, Sashikumaar Ganesan

TL;DR

FastVPINNs tackles the computational bottleneck of hp-VPINNs when solving PDEs on complex geometries by introducing tensor-based, mapped FE concepts and BLAS-friendly formulations. The method replaces per-element loops with preassembled premultipliers and a tensorized loss computation, enabling a single backpropagation pass even for skewed meshes. It demonstrates a 100× speedup over hp-VPINNs while achieving comparable or superior accuracy, including high-frequency solutions and inverse problems, on domains ranging from unit squares to gear-shaped geometries. This approach broadens the practical applicability of SciML for complex, real-world PDEs by combining variational formulations with scalable, GPU-accelerated tensor operations. The framework also provides insights into hyperparameter choices and paves the way for extensions to triangular elements and 3D domains.

Abstract

Variational Physics-Informed Neural Networks (VPINNs) utilize a variational loss function to solve partial differential equations, mirroring Finite Element Analysis techniques. Traditional hp-VPINNs, while effective for high-frequency problems, are computationally intensive and scale poorly with increasing element counts, limiting their use in complex geometries. This work introduces FastVPINNs, a tensor-based advancement that significantly reduces computational overhead and improves scalability. Using optimized tensor operations, FastVPINNs achieve a 100-fold reduction in the median training time per epoch compared to traditional hp-VPINNs. With proper choice of hyperparameters, FastVPINNs surpass conventional PINNs in both speed and accuracy, especially in problems with high-frequency solutions. Demonstrated effectiveness in solving inverse problems on complex domains underscores FastVPINNs' potential for widespread application in scientific and engineering challenges, opening new avenues for practical implementations in scientific machine learning.

FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries

TL;DR

FastVPINNs tackles the computational bottleneck of hp-VPINNs when solving PDEs on complex geometries by introducing tensor-based, mapped FE concepts and BLAS-friendly formulations. The method replaces per-element loops with preassembled premultipliers and a tensorized loss computation, enabling a single backpropagation pass even for skewed meshes. It demonstrates a 100× speedup over hp-VPINNs while achieving comparable or superior accuracy, including high-frequency solutions and inverse problems, on domains ranging from unit squares to gear-shaped geometries. This approach broadens the practical applicability of SciML for complex, real-world PDEs by combining variational formulations with scalable, GPU-accelerated tensor operations. The framework also provides insights into hyperparameter choices and paves the way for extensions to triangular elements and 3D domains.

Abstract

Variational Physics-Informed Neural Networks (VPINNs) utilize a variational loss function to solve partial differential equations, mirroring Finite Element Analysis techniques. Traditional hp-VPINNs, while effective for high-frequency problems, are computationally intensive and scale poorly with increasing element counts, limiting their use in complex geometries. This work introduces FastVPINNs, a tensor-based advancement that significantly reduces computational overhead and improves scalability. Using optimized tensor operations, FastVPINNs achieve a 100-fold reduction in the median training time per epoch compared to traditional hp-VPINNs. With proper choice of hyperparameters, FastVPINNs surpass conventional PINNs in both speed and accuracy, especially in problems with high-frequency solutions. Demonstrated effectiveness in solving inverse problems on complex domains underscores FastVPINNs' potential for widespread application in scientific and engineering challenges, opening new avenues for practical implementations in scientific machine learning.
Paper Structure (30 sections, 28 equations, 17 figures, 1 table)

This paper contains 30 sections, 28 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Governing Equations
  4. Physics Informed Neural Networks
  5. hp-Variational Physics Informed Neural Network
  6. Implementation of hp-VPINNs
  7. Overview of Current hp-VPINNs Implementation
  8. Need for Efficient hp-VPINNs Implementation
  9. FastVPINNs Methodology
  10. Mapped hp-VPINNs
  11. Optimization I: Enhancing Efficiency with Matrix-Vector Product Reformulation
  12. Optimization II: Overcoming the Element Looping for regular elements
  13. FastVPINNs: Generalized algorithm for Complex Geometries
  14. Experimental Setup and Design
  15. Evaluation of FastVPINNs
  16. ...and 15 more sections

Figures (17)

  • Figure 1: Schematics of Variational PINNs for a 2D Poisson problem.
  • Figure 4: Bilinear transformation.
  • Figure 5: Quadrilateral meshes with elements having constant Jacobian.
  • Figure 6: FastVPINNs Tensor schematic representation for residual computation.
  • Figure 7: Exact solutions of the two-dimensional Poisson's equation for given test-cases.
  • ...and 12 more figures