Port-Hamiltonian systems (pHS) allow for a structure-preserving modeling of dynamical systems. Coupling pHS via linear relations between input and output defines an overall pHS, which is structure preserving. However, in multiphysics applications, some subsystems do not allow for a physical pHS description, as (a) this is not available or (b) too expensive. Here, data-driven approaches can be used to deliver a pHS for such subsystems, which can then be coupled to the other subsystems in a structure-preserving way. In this work, we derive a data-driven identification approach for port-Hamiltonian differential algebraic equation (DAE) systems. The approach uses input and state space data to estimate nonlinear effort functions of pH-DAEs. As underlying technique, we us (multi-task) Gaussian processes. This work thereby extends over the current state of the art, in which only port-Hamiltonian ordinary differential equation systems could be identified via Gaussian processes. We apply this approach successfully to two applications from network design and constrained multibody system dynamics, based on pH-DAE system of index one and three, respectively.

Simulation-based methods for statistical inference have evolved dramatically over the past 50 years, keeping pace with technological advancements. The field is undergoing a new revolution as it embraces the representational capacity of neural networks, optimisation libraries and graphics processing units for learning complex mappings between data and inferential targets. The resulting tools are amortised, in the sense that they allow rapid inference through fast feedforward operations. In this article we review recent progress in the context of point estimation, approximate Bayesian inference, summary-statistic construction, and likelihood approximation. We also cover software, and include a simple illustration to showcase the wide array of tools available for amortised inference and the benefits they offer over Markov chain Monte Carlo methods. The article concludes with an overview of relevant topics and an outlook on future research directions.

A reliable knowledge structure is a prerequisite for building effective adaptive learning systems and intelligent tutoring systems. Pursuing an explainable and trustworthy knowledge structure, we propose a method for constructing causal knowledge networks. This approach leverages Bayesian networks as a foundation and incorporates causal relationship analysis to derive a causal network. Additionally, we introduce a dependable knowledge-learning path recommendation technique built upon this framework, improving teaching and learning quality while maintaining transparency in the decision-making process.

Assume that we would like to estimate the expected value of a function $F$ with respect to a density $\pi$. We prove that if $\pi$ is close enough under KL divergence to another density $q$, an independent Metropolis sampler estimator that obtains samples from $\pi$ with proposal density $q$, enriched with a variance reduction computational strategy based on control variates, achieves smaller asymptotic variance than that of the crude Monte Carlo estimator. The control variates construction requires no extra computational effort but assumes that the expected value of $F$ under $q$ is analytically available. We illustrate this result by calculating the marginal likelihood in a linear regression model with prior-likelihood conflict and a non-conjugate prior. Furthermore, we propose an adaptive independent Metropolis algorithm that adapts the proposal density such that its KL divergence with the target is being reduced. We demonstrate its applicability in a Bayesian logistic and Gaussian process regression problems and we rigorously justify our asymptotic arguments under easily verifiable and essentially minimal conditions.

Artificial intelligence (AI) is currently based largely on black-box machine learning models which lack interpretability. The field of eXplainable AI (XAI) strives to address this major concern, being critical in high-stakes areas such as the finance, legal and health sectors. We present an approach to defining AI models and their interpretability based on category theory. For this we employ the notion of a compositional model, which sees a model in terms of formal string diagrams which capture its abstract structure together with its concrete implementation. This comprehensive view incorporates deterministic, probabilistic and quantum models. We compare a wide range of AI models as compositional models, including linear and rule-based models, (recurrent) neural networks, transformers, VAEs, and causal and DisCoCirc models. Next we give a definition of interpretation of a model in terms of its compositional structure, demonstrating how to analyse the interpretability of a model, and using this to clarify common themes in XAI. We find that what makes the standard 'intrinsically interpretable' models so transparent is brought out most clearly diagrammatically. This leads us to the more general notion of compositionally-interpretable (CI) models, which additionally include, for instance, causal, conceptual space, and DisCoCirc models. We next demonstrate the explainability benefits of CI models. Firstly, their compositional structure may allow the computation of other quantities of interest, and may facilitate inference from the model to the modelled phenomenon by matching its structure. Secondly, they allow for diagrammatic explanations for their behaviour, based on influence constraints, diagram surgery and rewrite explanations. Finally, we discuss many future directions for the approach, raising the question of how to learn such meaningfully structured models in practice.

Test-time augmentation (TTA) is a well-known technique employed during the testing phase of computer vision tasks. It involves aggregating multiple augmented versions of input data. Combining predictions using a simple average formulation is a common and straightforward approach after performing TTA. This paper introduces a novel framework for optimizing TTA, called BayTTA (Bayesian-based TTA), which is based on Bayesian Model Averaging (BMA). First, we generate a model list associated with different variations of the input data created through TTA. Then, we use BMA to combine model predictions weighted by their respective posterior probabilities. Such an approach allows one to take into account model uncertainty, and thus to enhance the predictive performance of the related machine learning or deep learning model. We evaluate the performance of BayTTA on various public data, including three medical image datasets comprising skin cancer, breast cancer, and chest X-ray images and two well-known gene editing datasets, CRISPOR and GUIDE-seq. Our experimental results indicate that BayTTA can be effectively integrated into state-of-the-art deep learning models used in medical image analysis as well as into some popular pre-trained CNN models such as VGG-16, MobileNetV2, DenseNet201, ResNet152V2, and InceptionRes-NetV2, leading to the enhancement in their accuracy and robustness performance.

Failures of fairness or robustness in machine learning predictive settings can be due to undesired dependencies between covariates, outcomes and auxiliary factors of variation. A common strategy to mitigate these failures is data balancing, which attempts to remove those undesired dependencies. In this work, we define conditions on the training distribution for data balancing to lead to fair or robust models. Our results display that, in many cases, the balanced distribution does not correspond to selectively removing the undesired dependencies in a causal graph of the task, leading to multiple failure modes and even interference with other mitigation techniques such as regularization. Overall, our results highlight the importance of taking the causal graph into account before performing data balancing.

Specifying a prior distribution is an essential part of solving Bayesian inverse problems. The prior encodes a belief on the nature of the solution and this regularizes the problem. In this article we completely characterize a Gaussian prior that encodes the belief that the solution is a structured tensor. We first define the notion of (A,b)-constrained tensors and show that they describe a large variety of different structures such as Hankel, circulant, triangular, symmetric, and so on. Then we completely characterize the Gaussian probability distribution of such tensors by specifying its mean vector and covariance matrix. Furthermore, explicit expressions are proved for the covariance matrix of tensors whose entries are invariant under a permutation. These results unlock a whole new class of priors for Bayesian inverse problems. We illustrate how new kernel functions can be designed and efficiently computed and apply our results on two particular Bayesian inverse problems: completing a Hankel matrix from a few noisy measurements and learning an image classifier of handwritten digits. The effectiveness of the proposed priors is demonstrated for both problems. All applications have been implemented as reactive Pluto notebooks in Julia.

In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible. While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii). The proposed methodology results in an efficient derivative-free sampler flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.

We study the finite-time convergence of TD learning with linear function approximation under Markovian sampling. Existing proofs for this setting either assume a projection step in the algorithm to simplify the analysis, or require a fairly intricate argument to ensure stability of the iterates. We ask: \textit{Is it possible to retain the simplicity of a projection-based analysis without actually performing a projection step in the algorithm?} Our main contribution is to show this is possible via a novel two-step argument. In the first step, we use induction to prove that under a standard choice of a constant step-size $\alpha$, the iterates generated by TD learning remain uniformly bounded in expectation. In the second step, we establish a recursion that mimics the steady-state dynamics of TD learning up to a bounded perturbation on the order of $O(\alpha^2)$ that captures the effect of Markovian sampling. Combining these pieces leads to an overall approach that considerably simplifies existing proofs. We conjecture that our inductive proof technique will find applications in the analyses of more complex stochastic approximation algorithms, and conclude by providing some examples of such applications.