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

TL;DR

The paper tackles the limitation that ML-based PDE identification often binds models to a fixed dimension and coordinate frame, hindering generalization. It introduces a coordinate-free, dimension-agnostic framework built on exterior calculus, creating an invariant feature basis and learning evolution operators that operate across arbitrary spaces. Through FitzHugh-Nagumo, Barkley, Patlak-Keller-Segel, Helmholtz, and Navier–Stokes-inspired experiments, the authors demonstrate accurate cross-domain predictions across dimensions, curvatures, and topologies, including transitions from 1D training to 2D/3D deployment. This spatially liberated approach holds promise for integrating heterogeneous experimental data and enabling transferable, geometry-agnostic PDE modeling with potential impact across scientific computing and AI-assisted simulation.

Abstract

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.

Figures (14)

• Figure 1: We aim to demonstrate that it is possible to identify a class of evolutionary PDE systems in a coordinate- and dimension-free manner using neural networks. We package the input in a coordinate- and dimension-free manner. (A) We learn the one-dimensional space right-hand-side operator of the PDEs in a Cartesian coordinate. (B) We can then simulate the dynamics in arbitrary coordinates and dimensions (the figure illustrates the case of two-dimensional curved space).
• Figure 2: The Patlak-Keller-Segel field evolution dynamics in a periodic one-dimensional space, with different coordinate systems. The initial field configurations are the same, up to an extrapolation when transitioning between different coordinate systems. (A) The Cartesian and non-Cartesian coordinate systems, using Eq. \ref{['x_to_y']} with different values of $L$ to convert between $x$ and $y$. (B1) The time progression of the fields when integrating Eq. \ref{['full_glory_non']} to generate the true dynamics Cartesian coordinate $x$. (B2) The time progression of the fields when using the trained neural network to generate the learned dynamics in the Cartesian coordinate $x$. (C) Transformation of Fig. (B1) into the non-Cartesian coordinate $y$. (D1) The time progression of the fields when using the trained neural network to generate the learned dynamics in the non-Cartesian coordinate $y$. (D2) Transformation of Fig. (D1) into the Cartesian coordinate $x$.
• Figure 3: The time evolution dynamics of FitzHugh-Nagumo and Barkley systems in higher-dimensional Euclidean spaces. We study the FitzHugh-Nagumo system on a two-dimensional domain with periodic boundary conditions under various coordinate representations in (A), and investigate the Barkley system in three-dimensional space subject to Neumann (no-flux) boundary conditions in (B). For (A), we present the time evolution of the true dynamics obtained by integrating Eq. \ref{['FHN']} and compare it with the dynamics generated using the learned model at four different time frames: (A1)$t=0$, (A2)$t=15$, (A3)$t=30$, (A4)$t=45$. Each time frame consists of four two-dimensional plots: the top row shows the true dynamics, while the bottom row displays the neural network predictions; the left column corresponds to $V$-field and the right column to $W$-field of the FitzHugh-Nagumo system. For (B), we present the time evolution of the true dynamics obtained by integrating Eq. \ref{['BM']} and compare it with the dynamics generated using the learned model at four different time frames: (B1)$t=0$, (B2)$t=1.5$, (B3)$t=3.0$, (B4)$t=4.5$. Every time frame consists of four three-dimensional plots (each illustrated with two $x_3$-slices, located at $x_3=\pm 35$): the top row shows the true dynamics, while the bottom row displays the neural network predictions; the left column corresponds to $U$-field and the right column to $V$-field of the Barkley system.
• Figure 4: Implicit Coordinate-Free Representation: Left: Solutions of the 1-D Helmloltz problem (harmonic oscillator) in the original variables. Middle: Embedding of the solutions to $\mathcal{B}$, with black arrows depicting the two PCA modes obtained from the data. Right: Embedding of one prediction of a randomly initialized PINN in $\mathcal{B}$ (yellow), along with its projection (red) on the previously learned PCA submanifold . The PDE loss tests whether the yellow and green representations agree, i.e. if the PINN solution resides on the learned submanifold.
• Figure 5: Magnitude of velocity field for simulated flow around cylinder (top) and integrated (learned, coordinate-free) Taylor-Green Vortex dynamics (bottom)