Why Solving PDEs via Tensor Factorization Matters for Manifold Learning

Intrinsic Green’s Learning (IGL) tackles high-dimensional data by modeling target functions as solutions to linear partial differential equations (PDEs). Instead of treating a model as a black-box mapping, IGL assumes the data lies on a low-dimensional manifold where the target function $u$ is governed by an operator $L$ and a source term $f$. By framing the learning problem as finding these components, the framework attempts to capture the underlying physics or geometry of the data rather than just interpolating points.

Breaking the Dimensional Collapse

Manifold learning often hits a wall with "dimensional collapse," where an encoder maps high-dimensional input into a collapsed, useless subspace. IGL solves this through a two-stage training architecture that separates coordinate discovery from source fitting. In Stage 1, an encoder $\Psi$ discovers a coordinate chart on the low-dimensional manifold. In Stage 2, the model solves for source weights and kernel scales.

By solving Stage 2 to optimality at each step of the outer Stage 1 loop, the gradient passed back to the encoder only reflects how well the current coordinates serve the fitting problem. This is a pragmatic design choice: it prevents the encoder from trying to optimize the entire complexity of the source term and the geometry simultaneously. Additionally, IGL uses a Group Lasso penalty and learnable gates to automatically discover the intrinsic dimension, driving inactive coordinates to zero during training.

Scaling via Fubini’s Theorem and Tensor Decomposition

The computational bottleneck for solving PDEs in high dimensions is usually the integration step. IGL bypasses this by using tensor decomposition (specifically CP decomposition) on both the source term and the Green’s function. This allows the framework to apply Fubini’s theorem, factorizing a $d$-dimensional integral into $d$ independent 1D integrals.

This structure provides specific inductive biases. The choice of operator $L$ dictates the behavior: the Laplacian enforces smoothness, the Helmholtz operator enforces locality, and fractional operators allow for long-range correlations. Because integration is a stable operation compared to differentiation, this approach offers a smoother inductive bias than Physics-Informed Neural Networks (PINNs), which rely on residual minimization. Furthermore, by allowing for tensor ranks $K > 1$, IGL generalizes Generalized Additive Models (GAMs), enabling the representation of non-additive interactions like products or XOR patterns that a simple additive model would miss.

The Practical Reality of Low-Rank Assumptions

Let’s be clear about the limitations: IGL relies on the assumption that the source term and Green’s function actually admit low-rank tensor factorizations. It is not a drop-in replacement for generic deep learning models; it is a specialized tool. If the underlying physics of your data are high-rank or do not follow a PDE structure, the model will fail to generalize because the architectural prior is too restrictive.

In practice, IGL is best positioned as a geometric regularizer or a parallel head—like the ResIGL architecture—rather than a standalone solver. It excels when you have reason to believe the data follows a structured manifold, but it won't outperform a standard MLP on unstructured, high-entropy noise. The real story here is that IGL trades off general-purpose flexibility for sample efficiency and interpretability in specific geometric contexts.