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arXiv AI

Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions

The paper introduces Cut-DeepONet, a two‑stage training framework that explicitly models discontinuities in partial differential equations by partitioning the domain into smooth subregions and representing discontinuities as boundaries in a higher‑dimensional space. This lifting strategy aligns the operator learning task with neural network inductive biases, allowing the model to avoid directly approximating discontinuities. Experiments on benchmark PDEs demonstrate that Cut‑DeepONet outperforms state‑of‑the‑art methods, even with low‑resolution data, while using fewer trainable parameters.

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arXiv Machine Learning
Aug 20

Multi-stage neural operator learning with application for convolutions

The paper introduces two multi-stage neural operator learning frameworks—Deep Collocation Neural Operator (DCNO) and Deep Galerkin Neural Operator (DGNO)—for efficiently computing convolution integrals. DCNO is a supervised method that iteratively refines operator approximations by learning residuals from data pairs, while DGNO is an unsupervised approach that uses the weak form of a PDE residual when the operator can be represented by a PDE. Both frameworks build basis operators across multiple training stages, yielding markedly higher accuracy than one-shot learning and achieving near machine‑precision results for convolution problems, with significant efficiency gains for repeated queries or parametric variations.

By Zhiping Mao, Zhenye Wen, Yong Zhang, Xiaofei Zhao
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arXiv Machine Learning
Jun 5

On the training of physics-informed neural operators for solving parametric partial differential equations

arXiv:2606. 06164v1 Announce Type: new Abstract: Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data.

By Nanxi Chen, Chuanjie Cui, Airong Chen, Sifan Wang, Rujin Ma
llmscomputer-visionbenchmarks
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Hugging Face Trending Papers
Jun 4

On the training of physics-informed neural operators for solving parametric partial differential equations

Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning.

llmscomputer-visionbenchmarks
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arXiv Machine Learning
Aug 28

Enforcing Dirichlet Boundary Conditions in Operator Learning

The paper introduces a neural operator architecture that inherently satisfies homogeneous Dirichlet boundary conditions by constraining each layer’s output to lie within the span of selected Dirichlet eigenfunctions of the Laplacian. This design works for any bounded domain with a Lipschitz boundary and any discretization, avoiding the restrictions of previous methods. The authors prove universal approximation for their architecture and demonstrate its effectiveness on Darcy flow and Helmholtz equation problems.

By Andrew M. Stuart, Margaret Trautner
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arXiv Machine Learning
Aug 11

Test-time Generalization for Physics through Neural Operator Splitting

arXiv:2602. 00884v2 Announce Type: replace Abstract: Neural operators have shown promise in learning solution maps of partial differential equations (PDEs), but they often struggle to generalize when test inputs lie outside the training distribution, such as novel initial conditions, unseen PDE coefficients or unseen physics.

By Louis Serrano, Jiequn Han, Edouard Oyallon, Shirley Ho, Rudy Morel
fine-tuningbenchmarks
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