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FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition

Milad Ramezankhani, Rishi Yash Parekh, Anirudh Deodhar, Dagnachew Birru

TL;DR

The paper introduces FB-HyDON, a parameter-efficient, physics-informed operator-learning framework that combines hypernetworks with finite-basis domain decomposition to learn solution operators for complex PDEs. It provides two main variants: FB-HyPINN for physics-informed domain decomposition and FB-HyDON for operator learning, with a dual-hypernetwork design and chunking to maintain a constant parameter count as domain granularity increases. Across benchmarks including a high-frequency harmonic oscillator, 1D Burgers' equation, and Allen-Cahn, FB-HyDON outperforms DeepONet, MDON, and HyperDeepONet while requiring far fewer parameters, and FB-HyPINN demonstrates competitive results with fewer trainable parameters than FBPINN. This approach enables accurate, scalable PDE solvers with reduced data and computation, particularly for highly nonlinear or high-frequency dynamics, by leveraging domain decomposition and hypernetwork-driven parameter efficiency.

Abstract

Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large datasets for effective training. While physics-informed operators offer a data-agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite Basis Physics-Informed HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite basis functions, FB-HyDON effectively mitigates the training limitations associated with existing physics-informed operator learning methods. We validated our approach on the high-frequency harmonic oscillator, Burgers' equation at different viscosity levels, and Allen-Cahn equation demonstrating substantial improvements over other operator learning models.

FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition

TL;DR

The paper introduces FB-HyDON, a parameter-efficient, physics-informed operator-learning framework that combines hypernetworks with finite-basis domain decomposition to learn solution operators for complex PDEs. It provides two main variants: FB-HyPINN for physics-informed domain decomposition and FB-HyDON for operator learning, with a dual-hypernetwork design and chunking to maintain a constant parameter count as domain granularity increases. Across benchmarks including a high-frequency harmonic oscillator, 1D Burgers' equation, and Allen-Cahn, FB-HyDON outperforms DeepONet, MDON, and HyperDeepONet while requiring far fewer parameters, and FB-HyPINN demonstrates competitive results with fewer trainable parameters than FBPINN. This approach enables accurate, scalable PDE solvers with reduced data and computation, particularly for highly nonlinear or high-frequency dynamics, by leveraging domain decomposition and hypernetwork-driven parameter efficiency.

Abstract

Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large datasets for effective training. While physics-informed operators offer a data-agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite Basis Physics-Informed HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite basis functions, FB-HyDON effectively mitigates the training limitations associated with existing physics-informed operator learning methods. We validated our approach on the high-frequency harmonic oscillator, Burgers' equation at different viscosity levels, and Allen-Cahn equation demonstrating substantial improvements over other operator learning models.
Paper Structure (12 sections, 21 equations, 6 figures, 7 tables)

This paper contains 12 sections, 21 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Methodology
  4. FB-HyPINN
  5. FB-HyDON
  6. Results
  7. Physics-informed learning with finite basis domain decomposition
  8. Learning solution operators with parameter-efficient FB-HyDON
  9. Conclusion
  10. Additional results
  11. Model implementation and experiments details
  12. Benchmark problems setting

Figures (6)

  • Figure 1: a) Visualization of a set of window functions for harmonic oscillator case where time domain is divided into 12 subdomains. Different colors represent different subdomains and all subdomains are associated with window function based on their bounds. X-axis represents the time domain and y-axis shows window function values. Horizontal bars on the top represent the length of subdomains. b) Visualization of domain decomposition for time-space domain of 1D Allen-Cahn example. Black box represents the problem domain and overlapping subdomains are shown with gray rectangles.
  • Figure 2: Schematic of proposed FB-HyDON model. The inputs to the hypernetworks consist of a chunk identifier and a task-specific set of variables (sensory observations for Operator hypernet and subdomain information for Domain hypernet.) The outputs of the hypernets are merged together via the Hadamard product to produce the weights of the Target net. Physics-informed losses are obtained at query points and used to train the hypernets' parameters.
  • Figure 3: Comparison of FBPINN and FB-HyPINN's predictions (top row) and absolute error (bottom row) for solving 1D Burger's equation with $\nu = 0.001/\pi$ via physics-informed learning with domain decomposition. Both model has 25 subdomains.
  • Figure 4: Comparison of operators' prediction for solving the high-frequency harmonic oscillator with $w_0 = 80$ via physics-informed training. Each subplot represents a unique initial condition pair $[u_0, v_0]$. Both DON and HDON failed to learn the solution operator, while FB-HyDON was able to capture the system's nonlinearities and accurately predict the solution leveraging its domain decomposition capabilities. 12 subdomains were used to train FB-HyDON.
  • Figure 5: Comparison of operators' prediction (top row) and absolute error (bottom row) for solving 1D Burger's equation with $\nu = 0.005$ via physics-informed training. DeepONet and its modified version produced the largest error while having more trainable parameters. FB-HyDON (ours) outperformed other models and generated the most accurate predictions.
  • ...and 1 more figures