[Paper Review] Domain Agnostic Fourier Neural Operators

Ning Liu, Siavash Jafarzadeh+1 more|arXiv (Cornell University)|Apr 30, 2023

Non-Destructive Testing Techniques4 citations

TL;DR

This paper proposes Domain Agnostic Fourier Neural Operators (DAFNO), a novel neural operator architecture that explicitly encodes irregular domain boundaries using a smoothed characteristic function, enabling efficient and accurate learning of PDE solution operators on complex, evolving geometries via Fast Fourier Transform (FFT). DAFNO achieves state-of-the-art performance on material modeling and airfoil simulation benchmarks and generalizes to unseen crack patterns and loading conditions with only one training crack sample.

ABSTRACT

Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling problems on rectangular domains. To lift such a restriction and permit FFT on irregular geometries as well as topology changes, we introduce domain agnostic Fourier neural operator (DAFNO), a novel neural operator architecture for learning surrogates with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of FNOs, and leverage FFT to achieve rapid computations, in such a way that the geometric information is explicitly encoded in the architecture. In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as compared to baseline neural operator models on two benchmark datasets of material modeling and airfoil simulation. To further demonstrate the capability and generalizability of DAFNO in handling complex domains with topology changes, we consider a brittle material fracture evolution problem. With only one training crack simulation sample, DAFNO has achieved generalizability to unseen loading scenarios and substantially different crack patterns from the trained scenario. Our code and data accompanying this paper are available at https://github.com/ningliu-iga/DAFNO.

Motivation & Objective

• Address the limitation of standard Fourier Neural Operators (FNOs) that require rectangular, uniformly gridded domains for FFT-based efficiency.
• Overcome the instability and inaccuracy of zero-padding or masking techniques when applied to irregular domains or topology changes.
• Enable generalization of learned operators to unseen domain geometries and topological evolutions, such as crack propagation with bifurcation.
• Develop a neural operator architecture that retains the computational efficiency of FFT while explicitly encoding geometric boundary information.
• Demonstrate the method’s robustness and expressivity across diverse scientific computing problems involving complex, evolving domains.

Proposed method

• Introduce a smoothed characteristic function to represent the domain boundary, which is embedded directly into the integral layer of FNOs.
• Modify the FNO layer to perform spectral convolutions only within the physical domain, cutting off non-physical interactions outside the domain.
• Preserve the convolutional form of the kernel to maintain computational efficiency via Fast Fourier Transform (FFT).
• Propose two variants: eDAFNO (explicit FNO-based) and iDAFNO (implicit FNO-based), balancing accuracy and efficiency.
• Apply boundary smoothening to mitigate Gibbs phenomenon and improve numerical stability at domain edges.
• Train the model end-to-end on function pairs mapping inputs to solution fields on irregular domains, with data augmentation using randomized sinusoidal fields to improve generalization.

Experimental results

Research questions

• RQ1 Can a Fourier neural operator be made robust to arbitrary, irregular domain geometries while preserving FFT efficiency?
• RQ2 How can geometric boundary information be explicitly encoded in a neural operator to improve generalization to unseen domain shapes and topological changes?
• RQ3 Can a single training sample of a crack propagation process enable accurate prediction of diverse crack patterns under different loading conditions?
• RQ4 How does the proposed method compare to standard FNOs with masking or padding in terms of stability and accuracy on evolving domains?
• RQ5 To what extent can the model generalize to new boundary conditions and domain topologies not seen during training?

Key findings

• DAFNO achieves state-of-the-art accuracy on benchmark datasets for hyperelastic material modeling and airfoil simulation, outperforming baseline neural operator models.
• With only one training sample of a crack propagation scenario under 4 MPa loading, eDAFNO generalizes to unseen loading magnitudes (2 MPa and 6 MPa), accurately predicting distinct crack patterns including bifurcation and rapid growth.
• eDAFNO maintains stable predictions over long simulation times, while standard FNO with mask becomes unstable early due to distribution shift in the mask.
• Relative error in the characteristic function and displacement fields remains low for eDAFNO across all test scenarios, indicating high fidelity in capturing evolving domain topology.
• iDAFNO shows superior computational efficiency and reduced overfitting with limited data, while eDAFNO achieves slightly higher accuracy.
• The boundary smoothening technique effectively mitigates Gibbs phenomenon and improves numerical stability at domain boundaries.