Intrinsic Green's Learning (IGL)

• IGL is a supervised framework that expresses targets as inverse PDE solutions using a learned source from low-dimensional manifold data.
• It leverages low-rank tensor decompositions of both the source and Green’s kernel to collapse high-dimensional integrals into independent 1D integrals.
• The two-stage optimization, which separates coordinate discovery from source fitting, improves efficiency and prevents latent collapse.

Intrinsic Green’s Learning (IGL) is a supervised-learning framework that recasts prediction on a low-dimensional data manifold as an inverse PDE problem. Rather than learning the target function $u$ directly, IGL learns a source term $f$ and represents the target as the PDE solution obtained by integrating that source 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 (Quemy, 8 Jul 2026).

1. Core formulation

IGL assumes data lying on a manifold

$$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$

An encoder

$\Psi:\mathbb{R}^D \to \mathbb{R}^d$

maps ambient inputs $x$ to intrinsic coordinates

$\xi = \Psi(x)\in\mathbb{R}^d.$

The supervised target $u:\mathcal{M}\to\mathbb{R}^C$ is modeled as the solution of a linear PDE

$Lu = f,$

where $L$ is a chosen differential operator and $f$ is a learned source term. Using the Green’s function $f$0 of $f$1, the solution is written in integral form as

$f$2

where $f$3 is the learned intrinsic domain (Quemy, 8 Jul 2026).

The PDE structure is presented as an architectural prior, not a physical claim. In that formulation, the role of the PDE is to provide smoothness, linearity in the source coefficients, and a natural route to tensor factorization. This distinguishes IGL from direct function approximation: the learned object is the source term, while prediction is produced through a Green’s-function inverse solve rather than a direct regression map. A plausible implication is that the method places most of its inductive bias in the choice of coordinates and operator structure rather than in unrestricted function fitting.

2. Low-rank Green’s representation in intrinsic coordinates

IGL assumes that, in learned coordinates, the source is approximately CP-separable: $f$4 where $f$5 is the source tensor rank, $f$6 are learned weights, and $f$7 are univariate basis functions. The Green’s kernel is likewise approximated by a rank-$f$8 separated expansion: $f$9 with weights $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$0. Substituting both decompositions into the convolution and applying Fubini’s theorem yields

$$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$1

where each factor is one-dimensional: $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$2 The original $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$3-dimensional integral therefore collapses into $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$4 independent 1D integrals, and the overall evaluation cost scales linearly in the intrinsic dimension (Quemy, 8 Jul 2026).

The paper also gives a spectral version. If $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$5 is self-adjoint with eigenpairs $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$6, then

$$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$7

On product domains with separable operators $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$8, the eigenfunctions factor: $$\mathcal{M} \subset [0,1]^D,\qquad \dim(\mathcal{M}) = d \ll D.$$9 To handle the reciprocal of the sum of eigenvalues, IGL uses the exponential-sum trick: $\Psi:\mathbb{R}^D \to \mathbb{R}^d$0 This restores factorization in the spectral representation. In the paper’s account, this is the mechanism that reduces the high-dimensional inverse PDE solve to low-rank tensor operations (Quemy, 8 Jul 2026).

3. Two-stage optimization and intrinsic dimension discovery

A central feature of IGL is the separation of geometry discovery from source fitting. Stage 1 is coordinate or chart discovery: the encoder $\Psi:\mathbb{R}^D \to \mathbb{R}^d$1 learns a chart in which the target becomes low-rank and the PDE operator is approximately separable. The chart is not required to be isometric; any diffeomorphic chart that simplifies the tensor structure is acceptable. In practice, this stage updates encoder weights $\Psi:\mathbb{R}^D \to \mathbb{R}^d$2, coordinate gates $\Psi:\mathbb{R}^D \to \mathbb{R}^d$3, source anchor positions $\Psi:\mathbb{R}^D \to \mathbb{R}^d$4, kernel scale parameters $\Psi:\mathbb{R}^D \to \mathbb{R}^d$5, and rank gates $\Psi:\mathbb{R}^D \to \mathbb{R}^d$6. The outer objective is the reduced loss

$\Psi:\mathbb{R}^D \to \mathbb{R}^d$7

where $\Psi:\mathbb{R}^D \to \mathbb{R}^d$8 is the optimal inner solution (Quemy, 8 Jul 2026).

Stage 2 fits the source by a near-convex solve. With coordinates fixed, the source weights are obtained by least squares: $\Psi:\mathbb{R}^D \to \mathbb{R}^d$9 The design matrix is

$x$0

The paper characterizes this as the near-convex part: once the coordinates and kernel structure are fixed, the problem is linear in the source weights. The outer loop updates the encoder using the residual after this inner solve, and the envelope theorem is invoked to argue that gradients through the reduced objective reflect only how good the coordinates are, not how the source weights might compensate. This is the mechanism claimed to prevent “dimensional collapse” (Quemy, 8 Jul 2026).

IGL also includes coordinate gates $x$1, one per latent dimension. These gates are multiplied into coordinates and regularized with a group-lasso-style penalty. The architecture table explicitly notes that $x$2 are learned, that effective dimension $x$3 is discovered by group lasso on $x$4, and that effective source rank $x$5 is discovered by softplus gating on $x$6. Dimension selection is therefore embedded in the optimization rather than handled as a separate preprocessing step. This suggests that IGL treats intrinsic dimensionality as a learnable structural variable rather than a fixed hyperparameter.

4. Theoretical structure, assumptions, and complexity

The theoretical portion of IGL is organized around several structural claims. First, the factorized integral proof states that the CP form of the source plus the separated Green’s kernel implies the product-of-1D-integrals formula via Fubini’s theorem. Second, the spectral factorization proof states that the spectral Green’s representation combined with exponential-sum approximation yields a separable approximation. Third, the paper identifies a separability barrier on manifolds: on curved manifolds, the volume factor $x$7 can break separability. The proposed fix is to learn a compensated source $x$8, absorbing the metric term into the learned source. Fourth, the paper gives a mixed-smoothness justification: low tensor rank is coordinate-dependent, and the encoder’s role is to find coordinates where the source has bounded mixed derivatives and is therefore tensor-separable. Fifth, for separable $x$9, applying $\xi = \Psi(x)\in\mathbb{R}^d.$0 to a rank-$\xi = \Psi(x)\in\mathbb{R}^d.$1 tensor raises rank at most to $\xi = \Psi(x)\in\mathbb{R}^d.$2, so the approach remains structured (Quemy, 8 Jul 2026).

The paper also addresses null-space handling. Because Green’s functions omit zero modes, the solution is augmented with polynomial terms: $\xi = \Psi(x)\in\mathbb{R}^d.$3 This handles harmonic or polynomial null-space components. The paper does not present a formal end-to-end generalization theorem, but it gives approximation-theoretic and identifiability-style arguments based on mixed smoothness, low-rank tensor approximation, and the reduced objective (Quemy, 8 Jul 2026).

The complexity summary gives

$\xi = \Psi(x)\in\mathbb{R}^d.$4

per sample, where $\xi = \Psi(x)\in\mathbb{R}^d.$5 is the encoder cost and $\xi = \Psi(x)\in\mathbb{R}^d.$6 is the IGL head cost. For the Green’s kernel approximation, the paper also states that

$\xi = \Psi(x)\in\mathbb{R}^d.$7

for suitable kernels, essentially independent of ambient dimension when the operator has a spectral gap or is otherwise amenable to low-rank exponential-sum approximation (Quemy, 8 Jul 2026). A plausible implication is that the computational bottleneck moves from ambient-dimensional integration to latent-coordinate discovery.

5. Empirical behavior on synthetic manifolds and MNIST

On synthetic manifolds such as the Swiss Roll, the paper reports that two-stage training clearly outperforms joint training. Joint training often collapses the latent space to 1D or a tangled cluster, whereas two-stage training recovers the correct 2D manifold topology. In the Swiss Roll example, two-stage training achieves more than $\xi = \Psi(x)\in\mathbb{R}^d.$8 lower MSE. In a rotated-plane regression problem in $\xi = \Psi(x)\in\mathbb{R}^d.$9, two-stage training exhibits a sharp sample-efficiency phase transition: between $u:\mathcal{M}\to\mathbb{R}^C$0 and $u:\mathcal{M}\to\mathbb{R}^C$1, all two-stage kernels exceed $u:\mathcal{M}\to\mathbb{R}^C$2, while jointly trained methods plateau much lower. The best spectral two-stage model reaches $u:\mathcal{M}\to\mathbb{R}^C$3 at $u:\mathcal{M}\to\mathbb{R}^C$4. The paper also reports that different operator choices recover roughly the same effective intrinsic dimension, typically 3–4 on the Swiss Roll, reflecting the true dimension plus some curvature overhead (Quemy, 8 Jul 2026).

The framework is also used to represent sharp, non-axis-aligned decision boundaries on synthetic rotated squares, checkerboards, and stripes by using Gabor-wavelet-style operators and multi-scale superposition. This is presented as evidence that IGL is not limited to smooth regression (Quemy, 8 Jul 2026).

On MNIST, IGL is used as a regularizer in an autoencoder latent space. The reported results are:

• AE+IGL (Two-stage): linear probe $u:\mathcal{M}\to\mathbb{R}^C$5; silhouette $u:\mathcal{M}\to\mathbb{R}^C$6; smoothness $u:\mathcal{M}\to\mathbb{R}^C$7; active dimensions $u:\mathcal{M}\to\mathbb{R}^C$8 out of $u:\mathcal{M}\to\mathbb{R}^C$9.
• AE+IGL (Joint): linear probe $Lu = f,$0; silhouette $Lu = f,$1; active dimensions $Lu = f,$2.
• AE+Contrastive: best silhouette $Lu = f,$3; but all $Lu = f,$4 dimensions remain active (Quemy, 8 Jul 2026).

The interpretation given in the paper is that IGL achieves near-optimal classification while also producing a more geometrically organized latent space than the plain autoencoder, and that the two-stage version gives the best balance between classification, smoothness, and dimension reduction. The recovered intrinsic dimension of about $Lu = f,$5 aligns with prior estimates for MNIST, cited in the paper as roughly $Lu = f,$6–$Lu = f,$7 (Quemy, 8 Jul 2026).

6. Relation to Green’s-function learning literature

IGL belongs to a broader line of work that learns Green’s functions or solution operators, but it adopts a distinct supervised-learning-on-manifolds viewpoint. Earlier work on time-dependent parabolic PDEs derived “the first theoretically rigorous scheme for learning the associated solution operator, which takes the form of a convolution with a Green’s function $Lu = f,$8,” using hierarchical low-rank structure, randomized numerical linear algebra, and a spacetime partition tailored to parabolic scaling; the resulting guarantee is a relative $Lu = f,$9-error of $L$0 with high probability using at most $L$1 input-output training pairs (Boullé et al., 2022). That work treats Green’s learning as recovery of the PDE’s intrinsic propagator from forcing-solution pairs, whereas IGL uses a chosen inverse PDE as an architectural prior for supervised targets on a manifold.

The doctoral thesis “Data-driven discovery of Green’s functions” shifts PDE discovery from “discovering a differential equation” to “discovering the linear input–output map” induced by the PDE, and introduces rational neural networks with trainable rational activation functions because rational functions may have poles and take arbitrarily large values, which is ideal for approximating functions with singularities such as Green’s functions (Boullé, 2022). BI-GreenNet learns Green’s functions directly from PDE structure by subtracting the fundamental solution, writing $L$2, and representing the smooth remainder by layer potentials; it is effective on bounded domains, unbounded exterior domains, and interface problems, and the learned kernel can be reused across many right-hand sides and boundary conditions (Lin et al., 2022). BIN-G, by contrast, emphasizes a reusable free-space or domain-independent Green’s function for elliptic PDEs, represented by an RBF kernel-based neural network and trained by combining a PDE residual loss, boundary-integral losses for prescribed test functions, and a normalization term (Negi et al., 2024). A later singularity-encoded approach introduces an augmented variable

$L$3

so that the network learns a smooth surrogate in a higher-dimensional embedding rather than the singular kernel directly, and then uses the projected Green’s function as a preconditioner or hybrid correction operator (Sun et al., 15 Sep 2025).

In that context, IGL is distinctive for combining four ideas in one framework: learning coordinates on the manifold rather than assuming them; representing the target as a PDE solution through a Green’s-function convolution; exploiting low-rank tensor decomposition to reduce a high-dimensional integral to a product of 1D integrals; and splitting optimization into chart discovery and source fitting, making the fitting step nearly convex and preventing latent collapse (Quemy, 8 Jul 2026). A common misconception is that IGL is simply “using a PDE”; the paper explicitly argues that its novelty is not just “using a PDE,” but this particular combination of intrinsic coordinates, Green’s-function representation, tensor factorization, and two-stage optimization. Another common misconception is that the learned PDE must be physically true of the data; the paper instead presents the PDE structure as an architectural prior.