Interpretability is central to trustworthy machine learning, yet existing metrics rarely quantify how effectively data support an interpretive representation. We propose Interpretive Efficiency, a normalized, task-aware functional that measures the fraction of task-relevant information transmitted through an interpretive channel. The definition is grounded in five axioms ensuring boundedness, Blackwell-style monotonicity, data-processing stability, admissible invariance, and asymptotic consistency. We relate the functional to mutual information and derive a local Fisher-geometric expansion, then establish asymptotic and finite-sample estimation guarantees using standard empirical-process tools. Experiments on controlled image and signal tasks demonstrate that the measure recovers theoretical orderings, exposes representational redundancy masked by accuracy, and correlates with robustness, making it a practical, theory-backed diagnostic for representation design.

We propose a unified information-geometric framework that formalizes understanding in learning as a trade-off between informativeness and geometric simplicity. An encoder ϕ is evaluated by U(ϕ): = I(ϕ(X);Y) βC(ϕ), where C(ϕ) penalizes curvature and intrinsic dimensionality, enforcing smooth, low-complexity manifolds. Under mild manifold and regularity assumptions, we derive non-asymptotic bounds showing that generalization error scales with intrinsic dimension while curvature controls approximation stability, directly linking geometry to sample efficiency. To operationalize this theory, we introduce the Varia-tional Geometric Information Bottleneck (V-GIB); a varia-tional estimator that unifies mutual-information compression and curvature regularization through tractable geometric proxies (Hutchinson trace, Jacobian norms, and local PCA). Experiments across synthetic manifolds, few-shot settings, and real-world datasets (Fashion-MNIST, CIFAR-10) reveal a robust information-geometry Pareto frontier, stable estimators, and substantial gains in interpretive efficiency. Notably, fractional-data experiments on CIFAR-10 confirm that curvature-aware encoders maintain predictive power under data scarcity, validating the predicted efficiency-curvature law. Overall, V-GIB provides a principled and measurable route to representations that are geometrically coherent, data-efficient, and aligned with human-understandable structure. Keywords: geometry of understanding; information bottleneck; curvature regularization; few-shot learning; mutual information; Hutchinson trace estimator; inter-pretability; human-machine alignment.

We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs approximate solutions to partial differential equations (PDEs) by minimizing residual losses over sampled collocation and boundary points. We formalize both operator-level and variational notions of consistency, proving that residual minimization in Sobolev norms leads to convergence in energy and uniform norms under mild regularity. Deterministic stability bounds quantify how bounded perturbations to the network outputs propagate through the full composite loss, while probabilistic concentration results via McDiarmid's inequality yield sample complexity guarantees for residual-based generalization. A unified generalization bound links residual consistency, projection error, and perturbation sensitivity. Empirical results on elliptic, parabolic, and nonlinear PDEs confirm the predictive accuracy of our theoretical bounds across regimes. The framework identifies key structural principles, such as operator coercivity, activation smoothness, and sampling admissibility, that underlie robust and generalizable PINN training, offering principled guidance for the design and analysis of PDE-informed learning systems.

We develop a principled framework for discovering causal structure in partial differential equations (PDEs) using physics-informed neural networks and counterfactual perturbations. Unlike classical residual minimization or sparse regression methods, our approach quantifies operator-level necessity through functional interventions on the governing dynamics. We introduce causal sensitivity indices and structural deviation metrics to assess the influence of candidate differential operators within neural surrogates. Theoretically, we prove exact recovery of the causal operator support under restricted isometry or mutual coherence conditions, with residual bounds guaranteeing identifiability. Empirically, we validate the framework on both synthetic and real-world datasets across climate dynamics, tumor diffusion, and ocean flows. Our method consistently recovers governing operators even under noise, redundancy, and data scarcity, outperforming standard PINNs and DeepONets in structural fidelity. This work positions causal PDE discovery as a tractable and interpretable inference task grounded in structural causal models and variational residual analysis.

We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A \approx P D Q$, where $P \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{k \times k}$, and $Q \in \mathbb{R}^{k \times n}$. The decomposition is defined variationally by minimizing a regularized Frobenius loss, allowing control over rank, sparsity, and conditioning. Unlike algebraic factorizations such as LU or SVD, it is computed by alternating minimization. We establish existence and perturbation stability of the solution and show that each update has complexity $\mathcal{O}(n^2k)$. Benchmarks against truncated SVD, CUR, and nonnegative matrix factorization show improved reconstruction accuracy on MovieLens, MNIST, Olivetti Faces, and gene expression matrices, particularly under sparsity and noise.

Understanding how changes in community parasite prevalence alter the rate and age distribution of severe malaria is essential for optimizing control efforts. Paton et al. assessed the incidence of pediatric severe malaria admissions from 13 hospitals in East Africa from 2006 to 2020 (see the Perspective by Taylor and Slutsker). Each 25% increase in community parasite prevalence shifted hospital admissions toward younger children. Low rates of lifetime infections appeared to confer some immunity to severe malaria in very young children. Children under the age of 5 years thus need to remain a focus of disease prevention for malaria control. Science , abj0089, this issue p. [926][1]; see also abk3443, p. [855][2] The relationship between community prevalence of Plasmodium falciparum and the burden of severe, life-threatening disease remains poorly defined. To examine the three most common severe malaria phenotypes from catchment populations across East Africa, we assembled a dataset of 6506 hospital admissions for malaria in children aged 3 months to 9 years from 2006 to 2020. Admissions were paired with data from community parasite infection surveys. A Bayesian procedure was used to calibrate uncertainties in exposure (parasite prevalence) and outcomes (severe malaria phenotypes). Each 25% increase in prevalence conferred a doubling of severe malaria admission rates. Severe malaria remains a burden predominantly among young children (3 to 59 months) across a wide range of community prevalence typical of East Africa. This study offers a quantitative framework for linking malaria parasite prevalence and severe disease outcomes in children. [1]: /lookup/doi/10.1126/science.abj0089 [2]: /lookup/doi/10.1126/science.abk3443