# Towards Coordinate- and Dimension-Agnostic Machine Learning for Partial Differential Equations

arXiv: https://arxiv.org/abs/2505.16549 · HTML: https://mlsanity.com/paper/2505.16549/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 61 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)
- [Any-dimensional equivariant neural networks](https://mlsanity.com/paper/2306.06327/) (2023)
- [Graph neural networks and non-commuting operators](https://mlsanity.com/paper/2411.04265/) (2024)
- [Tensor learning with orthogonal, Lorentz, and symplectic symmetries](https://mlsanity.com/paper/2406.01552/) (2024)
- [Data-Driven, ML-assisted Approaches to Problem Well-Posedness](https://mlsanity.com/paper/2503.19255/) (2025)

## Built on by (most cited first, 2 of 2)

- [Warm-starting PDE solvers with any-dimensional machine learning](https://mlsanity.com/paper/2609.38916/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

The machine learning methods for data-driven identification of partial differential equations (PDEs) are typically defined for a given number of spatial dimensions and a choice of coordinates the data have been collected in. This dependence prevents the learned evolution equation from generalizing to other spaces. In this work, we reformulate the problem in terms of coordinate- and dimension-independent representations, paving the way toward what we call ``spatially liberated" PDE learning. To this end, we employ a machine learning approach to predict the evolution of scalar field systems expressed in the formalism of exterior calculus, which is coordinate-free and immediately generalizes to arbitrary dimensions by construction. We demonstrate the performance of this approach in the FitzHugh-Nagumo and Barkley reaction-diffusion models, as well as the Patlak-Keller-Segel model informed by in-situ chemotactic bacteria observations. We provide extensive numerical experiments that demonstrate that our approach allows for seamless transitions across various spatial contexts. We show that the field dynamics learned in one space can be used to make accurate predictions in other spaces with different dimensions, coordinate systems, boundary conditions, and curvatures.