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In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.

In this paper, we develop an {\em epsilon admissible subsets} (EAS) model selection approach for performing group variable selection in the high-dimensional multivariate regression setting. This EAS strategy is designed to estimate a posterior-like, generalized fiducial distribution over a parsimonious class of models in the setting of correlated predictors and/or in the absence of a sparsity assumption. The effectiveness of our approach, to this end, is demonstrated empirically in simulation studies, and is compared to other state-of-the-art model/variable selection procedures. Furthermore, assuming a matrix-Normal linear model we show that the EAS strategy achieves {\em strong model selection consistency} in the high-dimensional setting if there does exist a sparse, true data generating set of predictors. In contrast to Bayesian approaches for model selection, our generalized fiducial approach completely avoids the problem of simultaneously having to specify arbitrary prior distributions for model parameters and penalize model complexity; our approach allows for inference directly on the model complexity. \textcolor{black}{Implementation of the method is illustrated through yeast data to identify significant cell-cycle regulating transcription factors.

We conduct a non asymptotic study of the Cross Validation (CV) estimate of the generalization risk for learning algorithms dedicated to extreme regions of the covariates space. In this Extreme Value Analysis context, the risk function measures the algorithm's error given that the norm of the input exceeds a high quantile. The main challenge within this framework is the negligible size of the extreme training sample with respect to the full sample size and the necessity to re-scale the risk function by a probability tending to zero. We open the road to a finite sample understanding of CV for extreme values by establishing two new results: an exponential probability bound on the \Kfold CV error and a polynomial probability bound on the leave-\textrm{p}-out CV. Our bounds are sharp in the sense that they match state-of-the-art guarantees for standard CV estimates while extending them to encompass a conditioning event of small probability. We illustrate the significance of our results regarding high dimensional classification in extreme regions via a Lasso-type logistic regression algorithm. The tightness of our bounds is investigated in numerical experiments.

We explore the probabilistic partition of unity network (PPOU-Net) model in the context of high-dimensional regression problems and propose a general framework focusing on adaptive dimensionality reduction. With the proposed framework, the target function is approximated by a mixture of experts model on a low-dimensional manifold, where each cluster is associated with a local fixed-degree polynomial. We present a training strategy that leverages the expectation maximization (EM) algorithm. During the training, we alternate between (i) applying gradient descent to update the DNN coefficients; and (ii) using closed-form formulae derived from the EM algorithm to update the mixture of experts model parameters. Under the probabilistic formulation, step (ii) admits the form of embarrassingly parallelizable weighted least-squares solves. The PPOU-Nets consistently outperform the baseline fully-connected neural networks of comparable sizes in numerical experiments of various data dimensions. We also explore the proposed model in applications of quantum computing, where the PPOU-Nets act as surrogate models for cost landscapes associated with variational quantum circuits.

The capacity to address counterfactual "what if" inquiries is crucial for understanding and making use of causal influences. Traditional counterfactual inference usually assumes a structural causal model is available. However, in practice, such a causal model is often unknown and may not be identifiable. This paper aims to perform reliable counterfactual inference based on the (learned) qualitative causal structure and observational data, without a given causal model or even directly estimating conditional distributions. We re-cast counterfactual reasoning as an extended quantile regression problem using neural networks. The approach is statistically more efficient than existing ones, and further makes it possible to develop the generalization ability of the estimated counterfactual outcome to unseen data and provide an upper bound on the generalization error. Experiment results on multiple datasets strongly support our theoretical claims.

The package fnets for the R language implements the suite of methodologies proposed by Barigozzi et al. (2022) for the network estimation and forecasting of high-dimensional time series under a factor-adjusted vector autoregressive model, which permits strong spatial and temporal correlations in the data. Additionally, we provide tools for visualising the networks underlying the time series data after adjusting for the presence of factors. The package also offers data-driven methods for selecting tuning parameters including the number of factors, vector autoregressive order and thresholds for estimating the edge sets of the networks of interest in time series analysis. We demonstrate various features of fnets on simulated datasets as well as real data on electricity prices.

Supervised learning problems with side information in the form of a network arise frequently in applications in genomics, proteomics and neuroscience. For example, in genetic applications, the network side information can accurately capture background biological information on the intricate relations among the relevant genes. In this paper, we initiate a study of Bayes optimal learning in high-dimensional linear regression with network side information. To this end, we first introduce a simple generative model (called the Reg-Graph model) which posits a joint distribution for the supervised data and the observed network through a common set of latent parameters. Next, we introduce an iterative algorithm based on Approximate Message Passing (AMP) which is provably Bayes optimal under very general conditions. In addition, we characterize the limiting mutual information between the latent signal and the data observed, and thus precisely quantify the statistical impact of the network side information. Finally, supporting numerical experiments suggest that the introduced algorithm has excellent performance in finite samples.

This paper presents a Spatial Re-parameterization (SpRe) method for the N:M sparsity in CNNs. SpRe is stemmed from an observation regarding the restricted variety in spatial sparsity present in N:M sparsity compared with unstructured sparsity. Particularly, N:M sparsity exhibits a fixed sparsity rate within the spatial domains due to its distinctive pattern that mandates N non-zero components among M successive weights in the input channel dimension of convolution filters. On the contrary, we observe that unstructured sparsity displays a substantial divergence in sparsity across the spatial domains, which we experimentally verified to be very crucial for its robust performance retention compared with N:M sparsity. Therefore, SpRe employs the spatial-sparsity distribution of unstructured sparsity to assign an extra branch in conjunction with the original N:M branch at training time, which allows the N:M sparse network to sustain a similar distribution of spatial sparsity with unstructured sparsity. During inference, the extra branch can be further re-parameterized into the main N:M branch, without exerting any distortion on the sparse pattern or additional computation costs. SpRe has achieved a commendable feat by matching the performance of N:M sparsity methods with state-of-the-art unstructured sparsity methods across various benchmarks. Code and models are anonymously available at \url{//github.com/zyxxmu/SpRe}.

In this paper, we propose a model averaging approach for addressing model uncertainty in the context of partial linear functional additive models. These models are designed to describe the relation between a response and mixed-types of predictors by incorporating both the parametric effect of scalar variables and the additive effect of a functional variable. The proposed model averaging scheme assigns weights to candidate models based on the minimization of a multi-fold cross-validation criterion. Furthermore, we establish the asymptotic optimality of the resulting estimator in terms of achieving the lowest possible square prediction error loss under model misspecification. Extensive simulation studies and an application to a near infrared spectra dataset are presented to support and illustrate our method.

Data transformations are essential for broad applicability of parametric regression models. However, for Bayesian analysis, joint inference of the transformation and model parameters typically involves restrictive parametric transformations or nonparametric representations that are computationally inefficient and cumbersome for implementation and theoretical analysis, which limits their usability in practice. This paper introduces a simple, general, and efficient strategy for joint posterior inference of an unknown transformation and all regression model parameters. The proposed approach directly targets the posterior distribution of the transformation by linking it with the marginal distributions of the independent and dependent variables, and then deploys a Bayesian nonparametric model via the Bayesian bootstrap. Crucially, this approach delivers (1) joint posterior consistency under general conditions, including multiple model misspecifications, and (2) efficient Monte Carlo (not Markov chain Monte Carlo) inference for the transformation and all parameters for important special cases. These tools apply across a variety of data domains, including real-valued, integer-valued, compactly-supported, and positive data. Simulation studies and an empirical application demonstrate the effectiveness and efficiency of this strategy for semiparametric Bayesian analysis with linear models, quantile regression, and Gaussian processes.