Multiphysics simulation, which models the interactions between multiple physical processes, and multi-component simulation of complex structures are critical in fields like nuclear and aerospace engineering. Previous studies often rely on numerical solvers or machine learning-based surrogate models to solve or accelerate these simulations. However, multiphysics simulations typically require integrating multiple specialized solvers-each responsible for evolving a specific physical process-into a coupled program, which introduces significant development challenges. Furthermore, no universal algorithm exists for multi-component simulations, which adds to the complexity. Here we propose compositional Multiphysics and Multi-component Simulation with Diffusion models (MultiSimDiff) to overcome these challenges. During diffusion-based training, MultiSimDiff learns energy functions modeling the conditional probability of one physical process/component conditioned on other processes/components. In inference, MultiSimDiff generates coupled multiphysics solutions and multi-component structures by sampling from the joint probability distribution, achieved by composing the learned energy functions in a structured way. We test our method in three tasks. In the reaction-diffusion and nuclear thermal coupling problems, MultiSimDiff successfully predicts the coupling solution using decoupled data, while the surrogate model fails in the more complex second problem. For the thermal and mechanical analysis of the prismatic fuel element, MultiSimDiff trained for single component prediction accurately predicts a larger structure with 64 components, reducing the relative error by 40.3% compared to the surrogate model.

A central goal of machine learning is generalization. While the No Free Lunch Theorem states that we cannot obtain theoretical guarantees for generalization without further assumptions, in practice we observe that simple models which explain the training data generalize best: a principle called Occam's razor. Despite the need for simple models, most current approaches in machine learning only minimize the training error, and at best indirectly promote simplicity through regularization or architecture design. Here, we draw a connection between Occam's razor and in-context learning: an emergent ability of certain sequence models like Transformers to learn at inference time from past observations in a sequence. In particular, we show that the next-token prediction loss used to train in-context learners is directly equivalent to a data compression technique called prequential coding, and that minimizing this loss amounts to jointly minimizing both the training error and the complexity of the model that was implicitly learned from context. Our theory and the empirical experiments we use to support it not only provide a normative account of in-context learning, but also elucidate the shortcomings of current in-context learning methods, suggesting ways in which they can be improved. We make our code available at https://github.com/3rdCore/PrequentialCode.

Recently, there has been an increasing interest in performing post-hoc uncertainty estimation about the predictions of pre-trained deep neural networks (DNNs). Given a pre-trained DNN via back-propagation, these methods enhance the original network by adding output confidence measures, such as error bars, without compromising its initial accuracy. In this context, we introduce a novel family of sparse variational Gaussian processes (GPs), where the posterior mean is fixed to any continuous function when using a universal kernel. Specifically, we fix the mean of this GP to the output of the pre-trained DNN, allowing our approach to effectively fit the GP's predictive variances to estimate the DNN prediction uncertainty. Our approach leverages variational inference (VI) for efficient stochastic optimization, with training costs that remain independent of the number of training points, scaling efficiently to large datasets such as ImageNet. The proposed method, called fixed mean GP (FMGP), is architecture-agnostic, relying solely on the pre-trained model's outputs to adjust the predictive variances. Experimental results demonstrate that FMGP improves both uncertainty estimation and computational efficiency when compared to state-of-the-art methods.

Logistic regression is a classical model for describing the probabilistic dependence of binary responses to multivariate covariates. We consider the predictive performance of the maximum likelihood estimator (MLE) for logistic regression, assessed in terms of logistic risk. We consider two questions: first, that of the existence of the MLE (which occurs when the dataset is not linearly separated), and second that of its accuracy when it exists. These properties depend on both the dimension of covariates and on the signal strength. In the case of Gaussian covariates and a well-specified logistic model, we obtain sharp non-asymptotic guarantees for the existence and excess logistic risk of the MLE. We then generalize these results in two ways: first, to non-Gaussian covariates satisfying a certain two-dimensional margin condition, and second to the general case of statistical learning with a possibly misspecified logistic model. Finally, we consider the case of a Bernoulli design, where the behavior of the MLE is highly sensitive to the parameter direction.

Dynamic Mode Decomposition (DMD) has received increasing research attention due to its capability to analyze and model complex dynamical systems. However, it faces challenges in computational efficiency, noise sensitivity, and difficulty adhering to physical laws, which negatively affect its performance. Addressing these issues, we present Online Physics-informed DMD (OPIDMD), a novel adaptation of DMD into a convex optimization framework. This approach not only ensures convergence to a unique global optimum, but also enhances the efficiency and accuracy of modeling dynamical systems in an online setting. Leveraging the Bayesian DMD framework, we propose a probabilistic interpretation of Physics-informed DMD (piDMD), examining the impact of physical constraints on the DMD linear operator. Further, we implement online proximal gradient descent and formulate specific algorithms to tackle problems with different physical constraints, enabling real-time solutions across various scenarios. Compared with existing algorithms such as Exact DMD, Online DMD, and piDMD, OPIDMD achieves the best prediction performance in short-term forecasting, e.g. an $R^2$ value of 0.991 for noisy Lorenz system. The proposed method employs a time-varying linear operator, offering a promising solution for the real-time simulation and control of complex dynamical systems.

Particle-based Bayesian inference methods by sampling from a partition-free target (posterior) distribution, e.g., Stein variational gradient descent (SVGD), have attracted significant attention. We propose a path-guided particle-based sampling~(PGPS) method based on a novel Log-weighted Shrinkage (LwS) density path linking an initial distribution to the target distribution. We propose to utilize a Neural network to learn a vector field motivated by the Fokker-Planck equation of the designed density path. Particles, initiated from the initial distribution, evolve according to the ordinary differential equation defined by the vector field. The distribution of these particles is guided along a density path from the initial distribution to the target distribution. The proposed LwS density path allows for an efficient search of modes of the target distribution while canonical methods fail. We theoretically analyze the Wasserstein distance of the distribution of the PGPS-generated samples and the target distribution due to approximation and discretization errors. Practically, the proposed PGPS-LwS method demonstrates higher Bayesian inference accuracy and better calibration ability in experiments conducted on both synthetic and real-world Bayesian learning tasks, compared to baselines, such as SVGD and Langevin dynamics, etc.

Relying on the representation power of neural networks, most recent works have often neglected several factors involved in haze degradation, such as transmission (the amount of light reaching an observer from a scene over distance) and atmospheric light. These factors are generally unknown, making dehazing problems ill-posed and creating inherent uncertainties. To account for such uncertainties and factors involved in haze degradation, we introduce a variational Bayesian framework for single image dehazing. We propose to take not only a clean image and but also transmission map as latent variables, the posterior distributions of which are parameterized by corresponding neural networks: dehazing and transmission networks, respectively. Based on a physical model for haze degradation, our variational Bayesian framework leads to a new objective function that encourages the cooperation between them, facilitating the joint training of and thereby boosting the performance of each other. In our framework, a dehazing network can estimate a clean image independently of a transmission map estimation during inference, introducing no overhead. Furthermore, our model-agnostic framework can be seamlessly incorporated with other existing dehazing networks, greatly enhancing the performance consistently across datasets and models.

The design space of discrete-space diffusion or flow generative models are significantly less well-understood than their continuous-space counterparts, with many works focusing only on a simple masked construction. In this work, we aim to take a holistic approach to the construction of discrete generative models based on continuous-time Markov chains, and for the first time, allow the use of arbitrary discrete probability paths, or colloquially, corruption processes. Through the lens of optimizing the symmetric kinetic energy, we propose velocity formulas that can be applied to any given probability path, completely decoupling the probability and velocity, and giving the user the freedom to specify any desirable probability path based on expert knowledge specific to the data domain. Furthermore, we find that a special construction of mixture probability paths optimizes the symmetric kinetic energy for the discrete case. We find that we can outperform the mask construction even in text with kinetic-optimal mixture paths, while we can make use of domain-specific constructions of the probability path over the visual domain. Generative models over discrete spaces have not seen as much progress on the methodology side compared to continuous-space counterparts. For the most part, applications such as large language modeling rely solely on autoregressive models (Radford et al., 2019; Bommasani et al., 2021). The simplicity of autoregressive modeling has also motivated people to use them for multimodal generation, where other modalities, such as images and videos, are tokenized and modeled within an autoregressive framework (Van den Oord et al., 2016; Team, 2024; Sun et al., 2024). A promising framework that brings iterative refinement to the discrete case is to consider the use of Markov chains within a dynamical generative framework.

We study a system with finitely many groups of multi-action bandit processes, each of which is a Markov decision process (MDP) with finite state and action spaces and potentially different transition matrices when taking different actions. The bandit processes of the same group share the same state and action spaces and, given the same action that is taken, the same transition matrix. All the bandit processes across various groups are subject to multiple weakly coupled constraints over their state and action variables. Unlike the past studies that focused on the offline case, we consider the online case without assuming full knowledge of transition matrices and reward functions a priori and propose an effective scheme that enables simultaneous learning and control. We prove the convergence of the relevant processes in both the timeline and the number of the bandit processes, referred to as the convergence in the time and the magnitude dimensions. Moreover, we prove that the relevant processes converge exponentially fast in the magnitude dimension, leading to exponentially diminishing performance deviation between the proposed online algorithms and offline optimality. Jing Fu is with Department of Electrical and Electronic Engineering, School of Engineering, STEM College, RMIT University, Australia (e-mail: jing.fu@rmit.edu.au). Bill Moran is with Department of Electrical and Electronic Engineering, the University of Melbourne, VIC 3010, Australia (e-mail:wmoran@unimelb.edu.au).

Transformer-based architectures have achieved unprecedented success in time series analysis. However, facing the challenge of across-domain modeling, existing studies utilize statistical prior as prompt engineering fails under the huge distribution shift among various domains. In this paper, a Unified Time Series Diffusion (UTSD) model is established for the first time to model the multi-domain probability distribution, utilizing the powerful probability distribution modeling ability of Diffusion. Unlike the autoregressive models that capture the conditional probabilities of the prediction horizon to the historical sequence, we use a diffusion denoising process to model the mixture distribution of the cross-domain data and generate the prediction sequence for the target domain directly utilizing conditional sampling. The proposed UTSD contains three pivotal designs: (1) The condition network captures the multi-scale fluctuation patterns from the observation sequence, which are utilized as context representations to guide the denoising network to generate the prediction sequence; (2) Adapter-based fine-tuning strategy, the multi-domain universal representation learned in the pretraining stage is utilized for downstream tasks in target domains; (3) The diffusion and denoising process on the actual sequence space, combined with the improved classifier free guidance as the conditional generation strategy, greatly improves the stability and accuracy of the downstream task. We conduct extensive experiments on mainstream benchmarks, and the pre-trained UTSD outperforms existing foundation models on all data domains, exhibiting superior zero-shot generalization ability. After training from scratch, UTSD achieves comparable performance against domain-specific proprietary models. The empirical results validate the potential of UTSD as a time series foundational model.