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POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solution accuracy. This decomposition, combined with the constructive nature of the proofs, allows us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.

Factor analysis provides a canonical framework for imposing lower-dimensional structure such as sparse covariance in high-dimensional data. High-dimensional data on the same set of variables are often collected under different conditions, for instance in reproducing studies across research groups. In such cases, it is natural to seek to learn the shared versus condition-specific structure. Existing hierarchical extensions of factor analysis have been proposed, but face practical issues including identifiability problems. To address these shortcomings, we propose a class of SUbspace Factor Analysis (SUFA) models, which characterize variation across groups at the level of a lower-dimensional subspace. We prove that the proposed class of SUFA models lead to identifiability of the shared versus group-specific components of the covariance, and study their posterior contraction properties. Taking a Bayesian approach, these contributions are developed alongside efficient posterior computation algorithms. Our sampler fully integrates out latent variables, is easily parallelizable and has complexity that does not depend on sample size. We illustrate the methods through application to integration of multiple gene expression datasets relevant to immunology.

Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured data. The formulation is powerful and widely used in many sciences and engineering fields. However, one crucial assumption is that the unknown parameter must be deterministic. In reality, however, many problems are stochastic in nature, and the unknown parameter is random. The challenge then becomes recovering the full distribution of this unknown random parameter. It is a much more complex task. In this paper, we examine this problem in a general setting. In particular, we conceptualize the PDE solver as a push-forward map that pushes the parameter distribution to the generated data distribution. This way, the SDE-constrained optimization translates to minimizing the distance between the generated distribution and the measurement distribution. We then formulate a gradient-flow equation to seek the ground-truth parameter probability distribution. This opens up a new paradigm for extending many techniques in PDE-constrained optimization to that for systems with stochasticity.

In this paper, we present a comprehensive study on the convergence properties of Adam-family methods for nonsmooth optimization, especially in the training of nonsmooth neural networks. We introduce a novel two-timescale framework that adopts a two-timescale updating scheme, and prove its convergence properties under mild assumptions. Our proposed framework encompasses various popular Adam-family methods, providing convergence guarantees for these methods in training nonsmooth neural networks. Furthermore, we develop stochastic subgradient methods that incorporate gradient clipping techniques for training nonsmooth neural networks with heavy-tailed noise. Through our framework, we show that our proposed methods converge even when the evaluation noises are only assumed to be integrable. Extensive numerical experiments demonstrate the high efficiency and robustness of our proposed methods.

Neural network verification mainly focuses on local robustness properties. However, often it is important to know whether a given property holds globally for the whole input domain, and if not then for what proportion of the input the property is true. While exact preimage generation can construct an equivalent representation of neural networks that can aid such (quantitative) global robustness verification, it is intractable at scale. In this work, we propose an efficient and practical anytime algorithm for generating symbolic under-approximations of the preimage of neural networks based on linear relaxation. Our algorithm iteratively minimizes the volume approximation error by partitioning the input region into subregions, where the neural network relaxation bounds become tighter. We further employ sampling and differentiable approximations to the volume in order to prioritize regions to split and optimize the parameters of the relaxation, leading to faster improvement and more compact under-approximations. Evaluation results demonstrate that our approach is able to generate preimage approximations significantly faster than exact methods and scales to neural network controllers for which exact preimage generation is intractable. We also demonstrate an application of our approach to quantitative global verification.

We present a neural network-based simulation super-resolution framework that can efficiently and realistically enhance a facial performance produced by a low-cost, realtime physics-based simulation to a level of detail that closely approximates that of a reference-quality off-line simulator with much higher resolution (26x element count in our examples) and accurate physical modeling. Our approach is rooted in our ability to construct - via simulation - a training set of paired frames, from the low- and high-resolution simulators respectively, that are in semantic correspondence with each other. We use face animation as an exemplar of such a simulation domain, where creating this semantic congruence is achieved by simply dialing in the same muscle actuation controls and skeletal pose in the two simulators. Our proposed neural network super-resolution framework generalizes from this training set to unseen expressions, compensates for modeling discrepancies between the two simulations due to limited resolution or cost-cutting approximations in the real-time variant, and does not require any semantic descriptors or parameters to be provided as input, other than the result of the real-time simulation. We evaluate the efficacy of our pipeline on a variety of expressive performances and provide comparisons and ablation experiments for plausible variations and alternatives to our proposed scheme.

The purpose of this work is to present an improved energy conservation method for hyperelastodynamic contact problems based on specific normal compliance conditions. In order to determine this Improved Normal Compliance (INC) law, we use a Moreau--Yosida $\alpha$-regularization to approximate the unilateral contact law. Then, based on the work of Hauret--LeTallec \cite{hauret2006energy}, we propose in the discrete framework a specific approach allowing to respect the energy conservation of the system in adequacy with the continuous case. This strategy (INC) is characterized by a conserving behavior for frictionless impacts and admissible dissipation for friction phenomena while limiting penetration. Then, we detail the numerical treatment within the framework of the semi-smooth Newton method and primal-dual active set strategy for the normal compliance conditions with friction. We finally provide some numerical experiments to bring into light the energy conservation and the efficiency of the INC method by comparing with different classical methods from the literature throught representative contact problems.

This paper proposes two practical implementations of Four-Dimensional Variational (4D-Var) Ensemble Kalman Filter (4D-EnKF) methods for non-linear data assimilation. Our formulations' main idea is to avoid the intrinsic need for adjoint models in the context of 4D-Var optimization and, even more, to handle non-linear observation operators during the assimilation of observations. The proposed methods work as follows: snapshots of an ensemble of model realizations are taken at observation times, these snapshots are employed to build control spaces onto which analysis increments can be estimated. Via the linearization of observation operators at observation times, a line-search based optimization method is proposed to estimate optimal analysis increments. The convergence of this method is theoretically proven as long as the dimension of control-spaces equals model one. In the first formulation, control spaces are given by full-rank square root approximations of background error covariance matrices via the Bickel and Levina precision matrix estimator. In this context, we propose an iterative Woodbury matrix formula to perform the optimization steps efficiently. The last formulation can be considered as an extension of the Maximum Likelihood Ensemble Filter to the 4D-Var context. This employs pseudo-square root approximations of prior error covariance matrices to build control spaces. Experimental tests are performed by using the Lorenz 96 model. The results reveal that, in terms of Root-Mean-Square-Error values, both methods can obtain reasonable estimates of posterior error modes in the 4D-Var optimization problem. Moreover, the accuracies of the proposed filter implementations can be improved as the ensemble sizes are increased.

To generalize across tasks, an agent should acquire knowledge from past tasks that facilitate adaptation and exploration in future tasks. We focus on the problem of in-context adaptation and exploration, where an agent only relies on context, i.e., history of states, actions and/or rewards, rather than gradient-based updates. Posterior sampling (extension of Thompson sampling) is a promising approach, but it requires Bayesian inference and dynamic programming, which often involve unknowns (e.g., a prior) and costly computations. To address these difficulties, we use a transformer to learn an inference process from training tasks and consider a hypothesis space of partial models, represented as small Markov decision processes that are cheap for dynamic programming. In our version of the Symbolic Alchemy benchmark, our method's adaptation speed and exploration-exploitation balance approach those of an exact posterior sampling oracle. We also show that even though partial models exclude relevant information from the environment, they can nevertheless lead to good policies.

This work is dedicated to the study of how uncertainty estimation of the human motion prediction can be embedded into constrained optimization techniques, such as Model Predictive Control (MPC) for the social robot navigation. We propose several cost objectives and constraint functions obtained from the uncertainty of predicting pedestrian positions and related to the probability of the collision that can be applied to the MPC, and all the different variants are compared in challenging scenes with multiple agents. The main question this paper tries to answer is: what are the most important uncertainty-based criteria for social MPC? For that, we evaluate the proposed approaches with several social navigation metrics in an extensive set of scenarios of different complexity in reproducible synthetic environments. The main outcome of our study is a foundation for a practical guide on when and how to use uncertainty-aware approaches for social robot navigation in practice and what are the most effective criteria.