# A Physics-informed Multi-resolution Neural Operator

arXiv: https://arxiv.org/abs/2510.23810 · HTML: https://mlsanity.com/paper/2510.23810/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 71 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (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 the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learnings](https://mlsanity.com/paper/2107.11496/) (2021)
- [Time Resolution Independent Operator Learning](https://mlsanity.com/paper/2507.02524/) (2025)
- [Physics-informed Neural Networks for Heterogeneous Poroelastic Media](https://mlsanity.com/paper/2407.03372/) (2024)
- [A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains](https://mlsanity.com/paper/2409.05508/) (2024)
- [Learning Active Subspaces and Discovering Important Features with Gaussian Radial Basis Functions Neural Networks](https://mlsanity.com/paper/2307.05639/) (2023)

## Built on by (most cited first, 4 of 4)

- [Cheap2Rich: A Multi-Fidelity Framework for Data Assimilation and System Identification of Multiscale Physics -- Rotating Detonation Engines](https://mlsanity.com/paper/2601.20295/) (2026)
- [SENDAI: A Hierarchical Sparse-measurement, EfficieNt Data AssImilation Framework](https://mlsanity.com/paper/2601.21664/) (2026)
- [SpectraKAN: Conditioning Spectral Operators](https://mlsanity.com/paper/2602.05187/) (2026)
- [Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields](https://mlsanity.com/paper/2605.28551/) (2026)

## Abstract (arXiv)

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.