# Spectral-inspired Operator Learning with Limited Data and Unknown Physics

arXiv: https://arxiv.org/abs/2505.21573 · HTML: https://mlsanity.com/paper/2505.21573/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 55 references that are in the map.
- First posted on arXiv 27 May 2025.

## Builds on (most cited first, 30 of 30)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [BENO: Boundary-embedded Neural Operators for Elliptic PDEs](https://mlsanity.com/paper/2401.09323/) (2024)
- [A Neural PDE Solver with Temporal Stencil Modeling](https://mlsanity.com/paper/2302.08105/) (2023)
- [Deciphering and integrating invariants for neural operator learning with various physical mechanisms](https://mlsanity.com/paper/2311.14361/) (2023)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics](https://mlsanity.com/paper/2411.00040/) (2024)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition](https://mlsanity.com/paper/2302.10255/) (2023)
- [PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems](https://mlsanity.com/paper/2505.01736/) (2025)
- [Symbolic Neural Ordinary Differential Equations](https://mlsanity.com/paper/2503.08059/) (2025)

## Built on by (most cited first, 1 of 1)

- [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026)

## Abstract (arXiv)

Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability. To address these challenges, we propose Spectral-Inspired Neural Operator (SINO), which can model complex systems from just 2-5 trajectories, without requiring explicit PDE terms. Specifically, SINO automatically captures both local and global spatial derivatives from frequency indices, enabling a compact representation of the underlying differential operators in physics-agnostic regimes. To model nonlinear effects, it employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress aliasing. Extensive experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves state-of-the-art performance, with improvements of 1-2 orders of magnitude in accuracy. Particularly, with only 5 training trajectories, SINO outperforms data-driven methods trained on 1000 trajectories and remains predictive on challenging out-of-distribution cases where other methods fail.