Two general multi-stage neural operator learning frameworks applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training are introduced.
Abstract
Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.
This work introduces a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction, and develops this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces.
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FNO-Speed, an integrated solution incorporating the multi-level parallel FNO-aware mapping and tiling GEMM optimization strategy and the custom-sized high-frequency signal filtering scheme, is proposed, fully demonstrating the effectiveness of the FNO-Speed optimization strategy in improving FNO performance.
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Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.
Bai-Ming Zhang, Jin-Song Tang, Ying Xu et al.· 0 citations
Neural operators have emerged as a powerful paradigm for solving Partial Differential Equations. They learn mappings between infinite-dimensional function spaces and have been successfully applied in fields such as computational fluid dynamics and weather forecasting. However, existing methods face a fundamental dilemma. The Fourier Neural Operator is efficient at capturing global patterns but suffers from spectral leakage. Conversely, hybrid models combine spectral methods with standard convolutions but often introduce aliasing errors. This violates the critical property of resolution invariance. To address this, we propose the Spectral-Spatial Neural Operator. We introduce a dual-stream architecture that couples a spectral branch with an aliasing-free convolutional branch. This design allows the model to capture high-frequency residuals and sharp discontinuities without introducing grid-dependent artifacts. We conducted extensive experiments on the 1D Burgers, 2D Darcy Flow, and 2D Navier-Stokes equations. The results demonstrate that our method significantly outperforms state-of-the-art baselines in both prediction accuracy and zero-shot super-resolution stability.
Linfeng Liu· Poster Volume 0008 The 2026...· 0 citations
We introduce an amortized neural sampler that combines operator learning with flow methods for sampling. It maps SDE coefficient functions to pushforwards from a reference measure to the invariant measures, enabling efficient sampling across families of stochastic differential equations. Our framework shifts traditional sampling cost to an initial training phase, after which new SDE instances require only one encoder pass and a few ODE solver steps, independent of mixing time. To handle problems in high dimensions, we use Lagrangian trajectory sensors for the coefficient functions and cross attention in the architecture. We also theoretically establish the expressivity and resolution invariance of our framework. Experiments on 1D and 2D SDE families show competitive accuracy with substantial speedups over MCMC in regimes with slow mixing, transfer across sensor counts, and demonstration results on a 64D interacting particle SDE where traditional grid approaches are infeasible.
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We consider (feed-forward) Neural Networks (NNs) for the emulation of the solution to singularly perturbed second order boundary value problems, with two small parameters. We describe a shallow NN which exploits available asymptotic expansions for the solution. These additive decompositions into smooth and layer components, allow for derivative estimates which are explicit in the order of differentiation as well as the singular perturbation parameter(s) \cite{melenk, Irene, SX}. Utilizing such decompositions, we propose a simple NN for emulating the solution to such problems using the $\tanh$ activation function together with different training objectives, such as residual or energy minimization. The key idea is to augment the approximation space with suitable exponential functions, similar to enriched spaces in finite element methods, e.g.~\cite{Kellogg}. Numerical examples in one and two dimensions, including a smooth non-tensor-product domain, illustrate the resulting parameter-robust behavior over the tested perturbation ranges.
C. Xenophontos, Aayushman Raina· 0 citations
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