# Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE

arXiv: https://arxiv.org/abs/2607.07034 · HTML: https://mlsanity.com/paper/2607.07034/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 20 references that are in the map.
- First posted on arXiv 8 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [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)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Learning shape correspondence with anisotropic convolutional neural networks](https://mlsanity.com/paper/1605.06437/) (2016)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [The Training Process of Many Deep Networks Explores the Same Low-Dimensional Manifold](https://mlsanity.com/paper/2305.01604/) (2023)
- [Neural Eigenfunctions Are Structured Representation Learners](https://mlsanity.com/paper/2210.12637/) (2022)

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

- [Function Lives Where Variance Doesn't: Task-Weighted Charts of a Language Model's Computation](https://mlsanity.com/paper/2609.18989/) (2026)

## Abstract (arXiv)

We introduce Intrinsic Green's Learning (IGL), a framework that models a target function on a manifold as the solution to a linear PDE whose source term is learned from data. Rather than approximating the target directly, IGL learns a source and integrates it against a Green's kernel. An encoder discovers a low-dimensional coordinate chart on the manifold where both the source and the kernel decompose as low-rank tensors, collapsing a high-dimensional integral into independent one-dimensional integrals with cost linear in the intrinsic dimension. A two-stage algorithm separates coordinate discovery from source fitting, a near-convex linear solve, preventing the dimensional collapse of joint training. Learnable gates on each coordinate automatically discover the intrinsic dimension of the manifold. We validate IGL on synthetic manifolds and on MNIST, where it simultaneously achieves near-optimal classification and automatic recovery of the intrinsic dimension.