mlsanityData to 14 Oct 2026

Enabling Local Neural Operators to perform Equation-Free System-Level Analysis

arXiv 2505.02308 ↗First posted 5 May 2025cs.LG cs.NA math.DScited by 10

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Abstract (arXiv)

Neural Operators (NOs) provide a powerful framework for computations involving physical laws that can be modelled by (integro-) partial differential equations (PDEs), directly learning maps between infinite-dimensional function spaces that bypass both the explicit equation identification and their subsequent numerical solving. Still, NOs have so far primarily been employed to explore the dynamical behavior as surrogates of brute-force temporal simulations/predictions. Their potential for systematic rigorous numerical system-level tasks, such as fixed-point, stability, and bifurcation analysis - crucial for predicting irreversible transitions in real-world phenomena - remains largely unexplored. Toward this aim, inspired by the Equation-Free multiscale framework, we propose and implement a framework that integrates (local) NOs with advanced iterative numerical methods in the Krylov subspace, so as to perform efficient system-level stability and bifurcation analysis of large-scale dynamical systems. Beyond fixed point, stability, and bifurcation analysis enabled by local in time NOs, we also demonstrate the usefulness of local in space as well as in space-time ("patch") NOs in accelerating the computer-aided analysis of spatiotemporal dynamics. We illustrate our framework via three nonlinear PDE benchmarks: the 1D Allen-Cahn equation, which undergoes multiple concatenated pitchfork bifurcations; the Liouville-Bratu-Gelfand PDE, which features a saddle-node tipping point; and the FitzHugh-Nagumo (FHN) model, consisting of two coupled PDEs that exhibit both Hopf and saddle-node bifurcations.

Key facts. Cited by 10 papers in the map (data through 14 Oct 2026). Builds on 28 of its 108 references that are in the map. First posted on arXiv 5 May 2025.

Builds on

28 of 108 references.

References · 28

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2017FiLM: Visual Reasoning with a General Conditioning Layercited by 2,333
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
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  5. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
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  9. 2024Neural Operators with Localized Integral and Differential Kernelscited by 62
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  12. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 44
16 more
  1. 2021Learning Dissipative Dynamics in Chaotic Systemscited by 42
  2. 2019Coarse-scale PDEs from fine-scale observations via machine learningcited by 35
  3. 2021One-shot learning for solution operators of partial differential equationscited by 30
  4. 2022Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approachcited by 25
  5. 2023Training Neural Networks is NP-Hard in Fixed Dimensioncited by 23
  6. 2025Neural Chaos: A Spectral Stochastic Neural Operatorcited by 21
  7. 2022Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEscited by 19
  8. 2024Derivative-enhanced Deep Operator Networkcited by 19
  9. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16
  10. 2022Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approachcited by 15
  11. 2022Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysmscited by 14
  12. 2024Random Projection Neural Networks of Best Approximation: Convergence theory and practical applicationscited by 11
  13. 2023Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learningcited by 10
  14. 2022Local neural operator for solving transient partial differential equations on varied domainscited by 9
  15. 2025Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modelingcited by 8
  16. 2024Efficient Training of Deep Neural Operator Networks via Randomized Samplingcited by 6

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 5; 2026: 5.

10 papers.

  1. 2025Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problemscited by 4
  2. 2025A Physics-informed Multi-resolution Neural Operatorcited by 4
  3. 2025Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learningcited by 2
  4. 2025Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operatorscited by 1
  5. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  6. 2026RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservationcited by 0
  7. 2026Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approachcited by 0
  8. 2026Learning Contractive Integral Operators with Fredholm Integral Neural Operatorscited by 0
  9. 2026Fast Surrogate Modeling of Excitable and Oscillatory FitzHugh-Nagumo Dynamics with Parametric Neural Operatorscited by 0
  10. 2026Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamicscited by 0