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Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

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arxiv 2207.05209 v2 pith:M2WJS6PP submitted 2022-07-11 cs.LG cs.NAmath.NA

Zongyi Li , Daniel Zhengyu Huang , Burigede Liu , Anima Anandkumar This is my paper
classification cs.LGcs.NAmath.NA
keywords geo-fnopdesfouriergeometriessolversarbitrarycompareddesign
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abstract

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. LaDEEP: A Deep Learning-based Surrogate Model for Large Deformation of Elastic-Plastic Solids

    cs.LG 2025-06 conditional novelty 6.0 of 10

    A two-stage Transformer surrogate trained on finite element data predicts stretch-bending final shapes with about 0.17 mm mean absolute distance and over 10,000 times speedup versus FEM.

  2. Accelerating HEC-RAS: A Recurrent Neural Operator for Rapid River Forecasting

    cs.LG 2025-07 conditional novelty 5.0 of 10

    An autoregressive GRU-GeoFNO model predicts HEC-RAS stage and flow on 67 Mississippi reaches with a 3.45x speedup and a median absolute stage error of 0.31 feet on a year-long hold-out.

  3. GITO: Graph-Informed Transformer Operator for Learning Complex Partial Differential Equations

    cs.LG 2025-06 conditional novelty 5.0 of 10

    GITO, a graph-informed transformer operator, reports lower relative L2 errors than existing transformer-based neural operators on Navier-Stokes, heat conduction, and airfoil benchmark datasets.

  4. Latent Mamba Operator for Partial Differential Equations

    cs.LG 2025-05 conditional novelty 5.0 of 10

    LaMO replaces attention in latent-token neural operators with bidirectional state-space models and reports consistent accuracy gains on six PDE benchmarks.

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