Solutions to the Schrödinger bridge problem and its generalizations yield feedback control policies for optimal density steering over a controlled diffusion. To numerically compute the same, the dynamic Sinkhorn recursion has become a standard approach. The mathematical engine behind this approach is the Hopf-Cole transform that recasts the conditions for optimality into a system of boundary-coupled linear PDEs. Recent works pointed out that for the control-affine Schrödinger bridge problem, this exact linearity via Hopf-Cole transform, and thus the standard Sinkhorn recursion, apply only if the control and noise channels are proportional. When the channels do not match, the Hopf-Cole-transformed PDEs remain nonlinear, and no algorithm is available to solve the same. We advance the state-of-the-art by designing a Sinkhorn recursion with memory that leverages the structure of these nonlinear PDEs, and demonstrate how it solves the control-affine Schrödinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.

Only the constraints are presented here. Then, eq. 2 can be reformulated as follow: The complete optimal allocation of eq. 3 can be summarized by the following python script: """EF evaluation """ import copy import logging import os import cvxopt import numpy as np scalar = 10000 def cvxopt_solve_qp(P, q, G= None, h= None, **kwargs): P = 0.5 * (P + P.T) # make sure P is symmetric args = [cvxopt.matrix(P), The remaining two cases are additional edge cases related to the previous condition. The size and description of the dataset we used are presented in table. (see Table 6).

We formulate and solve the fixed horizon linear quadratic covariance steering problem in continuous time with a terminal cost measured in Hilbert-Schmidt (i.e., Frobenius) norm error between the desired and the controlled terminal covariances. For this problem, the necessary conditions of optimality become a coupled matrix ODE two-point boundary value problem. To solve this system of equations, we design a matricial recursive algorithm and prove its convergence. The proposed algorithm and its analysis make use of the linear fractional transforms parameterized by the state transition matrix of the associated Hamiltonian matrix. To illustrate the results, we provide two numerical examples: one with a two dimensional and another with a six dimensional state space.

Fair, transparent, and explainable decision-making remains a critical challenge in Olympic and Paralympic combat sports. This paper presents \emph{FST.ai 2.0}, an explainable AI ecosystem designed to support referees, coaches, and athletes in real time during Taekwondo competitions and training. The system integrates {pose-based action recognition} using graph convolutional networks (GCNs), {epistemic uncertainty modeling} through credal sets, and {explainability overlays} for visual decision support. A set of {interactive dashboards} enables human--AI collaboration in referee evaluation, athlete performance analysis, and Para-Taekwondo classification. Beyond automated scoring, FST.ai~2.0 incorporates modules for referee training, fairness monitoring, and policy-level analytics within the World Taekwondo ecosystem. Experimental validation on competition data demonstrates an {85\% reduction in decision review time} and {93\% referee trust} in AI-assisted decisions. The framework thus establishes a transparent and extensible pipeline for trustworthy, data-driven officiating and athlete assessment. By bridging real-time perception, explainable inference, and governance-aware design, FST.ai~2.0 represents a step toward equitable, accountable, and human-aligned AI in sports.

We greatly thank the reviewers for their constructive comments. However, in certain applications, there is scientific evidence for a parametric form of the triggering kernel (e.g., [Beggs We show here a small example with 5 nodes in Figure 1. We have similar observations for recovering other edges in this example. In each picture, the two blue curves outline the proposed CIs, and the two red curves outline the asymptotic CI. Moreover, we will adjust Section 2-3 as suggested.

The Neural Tangent Kernel (NTK) framework has provided deep insights into the training dynamics of neural networks under gradient flow. However, it relies on the assumption that the network is differentiable with respect to its parameters, an assumption that breaks down when considering non-smooth target functions or parameterized models exhibiting non-differentiable behavior. In this work, we propose a Nonlocal Neural Tangent Kernel (NNTK) that replaces the local gradient with a nonlocal interaction-based approximation in parameter space. Nonlocal gradients are known to exist for a wider class of functions than the standard gradient. This allows NTK theory to be extended to nonsmooth functions, stochastic estimators, and broader families of models. We explore both fixed-kernel and attention-based formulations of this nonlocal operator. We illustrate the new formulation with numerical studies.

In this paper, we formulate a mutual information optimal control problem (MIOCP) for discrete-time linear systems. This problem can be regarded as an extension of a maximum entropy optimal control problem (MEOCP). Differently from the MEOCP where the prior is fixed to the uniform distribution, the MIOCP optimizes the policy and prior simultaneously. As analytical results, under the policy and prior classes consisting of Gaussian distributions, we derive the optimal policy and prior of the MIOCP with the prior and policy fixed, respectively. Using the results, we propose an alternating minimization algorithm for the MIOCP. Through numerical experiments, we discuss how our proposed algorithm works.

Normalizing flows (NFs) provide uncorrelated samples from complex distributions, making them an appealing tool for parameter estimation. However, the practical utility of NFs remains limited by their tendency to collapse to a single mode of a multimodal distribution. In this study, we show that annealing with an adaptive schedule based on the effective sample size (ESS) can mitigate mode collapse. We demonstrate that our approach can converge the marginal likelihood for a biochemical oscillator model fit to time-series data in ten-fold less computation time than a widely used ensemble Markov chain Monte Carlo (MCMC) method. We show that the ESS can also be used to reduce variance by pruning the samples. We expect these developments to be of general use for sampling with NFs and discuss potential opportunities for further improvements.

Structural decoupling has played an essential role in model-based fault isolation and estimation in past decades, which facilitates accurate fault localization and reconstruction thanks to the diagonal transfer matrix design. However, traditional methods exhibit limited effectiveness in modeling high-dimensional nonlinearity and big data, and the decoupling idea has not been well-valued in data-driven frameworks. Known for big data and complex feature extraction capabilities, deep learning has recently been used to develop residual generation models. Nevertheless, it lacks decoupling-related diagnostic designs. To this end, this paper proposes a transfer learning-based input-output decoupled network (TDN) for diagnostic purposes, which consists of an input-output decoupled network (IDN) and a pre-trained variational autocoder (VAE). In IDN, uncorrelated residual variables are generated by diagonalization and parallel computing operations. During the transfer learning phase, knowledge of normal status is provided according to VAE's loss and maximum mean discrepancy loss to guide the training of IDN. After training, IDN learns the mapping from faulty to normal, thereby serving as the fault detection index and the estimated fault signal simultaneously. At last, the effectiveness of the developed TDN is verified by a numerical example and a chemical simulation.