Manifold-Informed Neural Networks for Low-Label ECG Electrolyte Inference

Physics-informed neural networks (PINNs) supervise on the residual of a known differential operator. Many high-stakes inference problems admit no such operator: clinical signals, behavioral measurements, materials properties. We introduce the Manifold-Informed Neural Network (MINN), a PINN-parallel paradigm that replaces the physics operator residual with a manifold operator residual: a discrete Poisson equation ${\mathbf{L}}_G\mathbf{H}=s_{\phi }(\mathbf{x},\mathbf{m})$ on a $k$-nearest-neighbor graph in the learned latent space, with a learned, task-aware source. The construction is a structural parallel to PINNs (squared operator residual plus label data, in place of operator residual plus boundary data); the $s=0$ limit recovers the homogeneous Laplace equation ${\mathbf{L}}_G\mathbf{H}=\mathbf{0}$ (classical manifold regularization solves the same equation but minimizes the Dirichlet energy $\mathrm{tr}({\mathbf{H}}^{\mathrm{\top }}{\mathbf{L}}_G\mathbf{H})$, a different penalty form). The source $s_{\phi }$ takes input features and auxiliary metadata directly, breaking the homogeneous smoothness prior that limits classical Laplacian penalties; gated Feature-wise Linear Modulation separately fuses metadata into the prediction branch. We evaluate MINN on three-class classification of potassium, calcium, and magnesium from 12-lead ECGs in MIMIC-IV-ECG, against twelve baselines spanning supervised and tabular methods, ECG foundation models, and manifold/graph approaches. MINN attains the highest K${}^{+}$ Macro-F1 of 0.61 (best baseline 0.55), with consistent gains on Ca${}^{2+}$ and Mg${}^{2+}$ and a 31% relative improvement at 1% labeled data. Six targeted ablations show that the inhomogeneous Poisson formulation, not graph smoothing alone, drives the gain.