TL;DR
This paper introduces a coordinate- and dimension-independent machine learning framework for PDEs using exterior calculus, enabling models to generalize across different spatial dimensions and coordinate systems.
Contribution
It reformulates PDE learning in a coordinate-free manner with exterior calculus, allowing seamless generalization across arbitrary dimensions and coordinate choices.
Findings
The approach generalizes PDE models to different spatial dimensions.
Models trained in one space predict accurately in other spaces.
Demonstrated on reaction-diffusion and chemotaxis models.
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…
Peer Reviews
Decision·Submitted to ICLR 2026
Originality: The paper introduces a geometric formulation of PDE learning that enforces coordinate and dimension invariance. While the mathematical tools are drawn from established invariant theory, the combination with neural PDE modeling provides a moderately original framing rather than a fundamentally new methodology. Quality: The experiments are competently executed on standard reaction–diffusion systems. The results qualitatively demonstrate that a model trained in one coordinate system c
Limited generality of the framework. Despite claiming coordinate- and dimension-agnostic PDE learning, the method is applicable only to PDEs involving first-order time derivatives and up to second-order spatial derivatives. This excludes many physically and scientifically important systems such as the Cahn–Hilliard, Kuramoto–Sivashinsky, Swift–Hohenberg, and wave equations. The paper briefly mentions that higher-order operators could be included by extending the invariant basis, but this is not
(i) Principled invariance: the paper builds a coordinate and dimension free feature library, so the learned law depends only on intrinsic geometric operators, and not on the chosen coordinates. A coordinate free integrator lets a model trained in simple 1D settings transfer directly to new coordinates, higher dimensions, and curved manifolds by recomputing the same invariants with the target metric, (ii) Cross domain demos: the authors show successful transfer 1D to 2D/3D, across coordinate syst
(i) Restricted operator class (up to second-order spatial derivatives): the invariant library B is built only from fields, Laplacians, and gradient inner products, (ii) The learned manifold in B space does not encode which boundary/initial data make the problem well posed, which must be supplied separately when moving to new domains, (iii) Limited stability analysis: long horizon stability or higher order integrators are not explored, (iv) The 2D Navier–Stokes example assumes that the pressure f
* (Technical contribution) The paper shows that the dynamics learned in one space can be used to make accurate predictions in other spaces with different dimensions, coordinate systems, boundary conditions, and curvatures. * (Technical contribution & Novelty) The paper push previous ideas further and enables PDE learning not only in a coordinate-independent way but also independently of the data domain, dimension, and geometry: a novel contribution.
* The proposed approach can be applied only to invariant systems under local orthogonal transformations and diffeomorphic translations. * How can we remove this resstriction? * The paper, Section 2 in particular, is hard to follow, requiring an extensive revision. * How can we compute the Laplacians and inner products of the gradients (lines 166-168) from the "given data" (line 150)? * I assume the "given data" are physically observable data and are the fields \\Psi_j. Is this correct?
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