arXiv subjects

Lennon Shikhman

Publications and source records attributed to Lennon Shikhman.

3 recordsLinked to original sources

arXiv2026

Kernel-Robust Dynamics for Reaction-Diffusion Equations with Measure-Valued Delay

We study reaction-diffusion equations with finite signed measure delays and nonlinear, possibly nonlocal feedback. We treat abstract reactions that are Lipschitz on bounded $L^2$-balls with quadratic coercivity, and pointwise cubic polynomials with positive leading coefficient in spatial dimensions at most three. For affine-growth feedback and time-independent forcing in $H^{-1}(\Omega)$, both classes generate global weak solution semiflows on $X=C([-r,0];L^2(\Omega))$. Besides total-variation Lipschitz stability, we obtain quantitative weak-star stability with modulus $\varpi(d)=d\log(e+d^{-1})$ for the abstract class and $\varpi(d)^{1/2}$ for cubic reactions, where $d$ is the bounded-Lipschitz distance between delay measures. The estimates are uniform on bounded sets of continuous histories and yield concentration and atomic-quadrature rates. A damping condition gives common compact absorption for the abstract class. Cubic coercivity removes that smallness condition on every fixed total-variation-bounded kernel class. In both cases the global attractors are upper semicontinuous in $X$ and in $C([-r,0];H_0^1(\Omega))$.

math.AP↗

arXiv2026

Diagnosing Failure Modes of Neural Operators Across Diverse PDE Families

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under structured shifts in coefficients, boundary conditions, discretization, and rollout horizon. Yet evaluation is still often dominated by in-distribution test error, making robustness difficult to assess. We introduce a standardized stress-testing framework for neural PDE solvers under deployment-relevant shift. We instantiate it on three representative architectures -- Fourier Neural Operators (FNOs), a DeepONet-style model, and convolutional neural operators (CNOs) -- across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Across 750 trained models, we measure robustness using baseline-normalized degradation factors together with spectral and rollout diagnostics. The resulting comparisons reveal that strong in-distribution accuracy does not reliably predict robustness, and that failure patterns depend jointly on architecture and PDE family. Our results provide a clearer basis for evaluating robustness claims in neural PDE solvers and suggest that function-space generalization under structured shift should be treated as a first-class evaluation target.

cs.LG↗

arXiv2026

Entropy Production in Machine Learning Under Fokker-Planck Probability Flow

Machine learning models deployed in nonstationary environments inevitably experience performance degradation due to data drift. While numerous drift detection heuristics exist, most lack a dynamical interpretation and provide limited guidance on how retraining decisions should be balanced against operational cost. In this work, we propose an entropy-based retraining framework grounded in nonequilibrium statistical physics. Interpreting drift as probability flow governed by a Fokker-Planck equation, we quantify model-data mismatch using relative entropy and show that its time derivative admits an entropy-balance decomposition featuring a nonnegative entropy production term driven by probability currents. Guided by this theory, we implement an entropy-triggered retraining policy using an exponentially weighted moving-average (EWMA) control statistic applied to a streaming kernel density estimator of the Kullback-Leibler divergence. We evaluate this approach across multiple nonstationary data streams. In synthetic, financial, and web-traffic domains, entropy-based retraining achieves predictive performance comparable to frequent retraining while reducing retraining frequency by one to two orders of magnitude. However, in a challenging biomedical ECG setting, the entropy-based trigger underperforms the maximum-frequency baseline, highlighting limitations of feature-space entropy monitoring under complex label-conditional drift.

cs.LG↗