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We investigate a convective Brinkman--Forchheimer problem coupled with a heat transfer equation. The investigated model considers thermal diffusion and viscosity depending on the temperature. We prove the existence of a solution without restriction on the data and uniqueness when the solution is slightly smoother and the data is suitably restricted. We propose a finite element discretization scheme for the considered model and derive convergence results and a priori error estimates. Finally, we illustrate the theory with numerical examples.

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference and stochastic calculus, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We show that these phase-transitions are always in a mean-field universality class, as they are the result of a self-consistency condition in the generative dynamics. We argue that the critical instability that arises from the phase transitions lies at the heart of their generative capabilities, which are characterized by a set of mean field critical exponents. Furthermore, using the statistical physics of disordered systems, we show that memorization can be understood as a form of critical condensation corresponding to a disordered phase transition. Finally, we show that the dynamic equation of the generative process can be interpreted as a stochastic adiabatic transformation that minimizes the free energy while keeping the system in thermal equilibrium.

We study general coordinate-wise MCMC schemes (such as Metropolis-within-Gibbs samplers), which are commonly used to fit Bayesian non-conjugate hierarchical models. We relate their convergence properties to the ones of the corresponding (potentially not implementable) Gibbs sampler through the notion of conditional conductance. This allows us to study the performances of popular Metropolis-within-Gibbs schemes for non-conjugate hierarchical models, in high-dimensional regimes where both number of datapoints and parameters increase. Given random data-generating assumptions, we establish dimension-free convergence results, which are in close accordance with numerical evidences. Applications to Bayesian models for binary regression with unknown hyperparameters and discretely observed diffusions are also discussed. Motivated by such statistical applications, auxiliary results of independent interest on approximate conductances and perturbation of Markov operators are provided.

In logistic regression modeling, Firth's modified estimator is widely used to address the issue of data separation, which results in the nonexistence of the maximum likelihood estimate. Firth's modified estimator can be formulated as a penalized maximum likelihood estimator in which Jeffreys' prior is adopted as the penalty term. Despite its widespread use in practice, the formal verification of the corresponding estimate's existence has not been established. In this study, we establish the existence theorem of Firth's modified estimate in binomial logistic regression models, assuming only the full column rankness of the design matrix. We also discuss other binomial regression models obtained through alternating link functions and prove the existence of similar penalized maximum likelihood estimates for such models.

Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under weak assumptions and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals, adding to the literature on confidence sequences (CS) -- sequences of confidence intervals that are uniformly valid over time -- which provide valid inference at arbitrary stopping times and incur no penalties for "peeking" at the data, unlike classical confidence intervals which require the sample size to be fixed in advance. Existing CSs in the literature are nonasymptotic, enjoying finite-sample guarantees but not the aforementioned broad applicability of asymptotic confidence intervals. This work provides a definition for "asymptotic CSs" and a general recipe for deriving them. Asymptotic CSs forgo nonasymptotic validity for CLT-like versatility and (asymptotic) time-uniform guarantees. While the CLT approximates the distribution of a sample average by that of a Gaussian for a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen) to uniformly approximate the entire sample average process by an implicit Gaussian process. As an illustration, we derive asymptotic CSs for the average treatment effect in observational studies (for which nonasymptotic bounds are essentially impossible to derive even in the fixed-time regime) as well as randomized experiments, enabling causal inference in sequential environments.

Randomized Controlled Trials (RCTs) may suffer from limited scope. In particular, samples may be unrepresentative: some RCTs over- or under- sample individuals with certain characteristics compared to the target population, for which one wants conclusions on treatment effectiveness. Re-weighting trial individuals to match the target population can improve the treatment effect estimation. In this work, we establish the exact expressions of the bias and variance of such reweighting procedures -- also called Inverse Propensity of Sampling Weighting (IPSW) -- in presence of categorical covariates for any sample size. Such results allow us to compare the theoretical performance of different versions of IPSW estimates. Besides, our results show how the performance (bias, variance, and quadratic risk) of IPSW estimates depends on the two sample sizes (RCT and target population). A by-product of our work is the proof of consistency of IPSW estimates. Results also reveal that IPSW performances are improved when the trial probability to be treated is estimated (rather than using its oracle counterpart). In addition, we study choice of variables: how including covariates that are not necessary for identifiability of the causal effect may impact the asymptotic variance. Including covariates that are shifted between the two samples but not treatment effect modifiers increases the variance while non-shifted but treatment effect modifiers do not. We illustrate all the takeaways in a didactic example, and on a semi-synthetic simulation inspired from critical care medicine.

With an aim to analyse the performance of Markov chain Monte Carlo (MCMC) methods, in our recent work we derive a large deviation principle (LDP) for the empirical measures of Metropolis-Hastings (MH) chains on a continuous state space. One of the (sufficient) assumptions for the LDP involves the existence of a particular type of Lyapunov function, and it was left as an open question whether or not such a function exists for specific choices of MH samplers. In this paper we analyse the properties of such Lyapunov functions and investigate their existence for some of the most popular choices of MCMC samplers built on MH dynamics: Independent Metropolis Hastings, Random Walk Metropolis, and the Metropolis-adjusted Langevin algorithm. We establish under what conditions such a Lyapunov function exists, and from this obtain LDPs for some instances of the MCMC algorithms under consideration. To the best of our knowledge, these are the first large deviation results for empirical measures associated with Metropolis-Hastings chains for specific choices of proposal and target distributions.

In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution strategy.

Application of Neural Networks to river hydraulics is fledgling, despite the field suffering from data scarcity, a challenge for machine learning techniques. Consequently, many purely data-driven Neural Networks proved to lack predictive capabilities. In this work, we propose to mitigate such problem by introducing physical information into the training phase. The idea is borrowed from Physics-Informed Neural Networks which have been recently proposed in other contexts. Physics-Informed Neural Networks embed physical information in the form of the residual of the Partial Differential Equations (PDEs) governing the phenomenon and, as such, are conceived as neural solvers, i.e. an alternative to traditional numerical solvers. Such approach is seldom suitable for environmental hydraulics, where epistemic uncertainties are large, and computing residuals of PDEs exhibits difficulties similar to those faced by classical numerical methods. Instead, we envisaged the employment of Neural Networks as neural operators, featuring physical constraints formulated without resorting to PDEs. The proposed novel methodology shares similarities with data augmentation and regularization. We show that incorporating such soft physical information can improve predictive capabilities.

Surrogate modelling techniques have seen growing attention in recent years when applied to both modelling and optimisation of industrial design problems. These techniques are highly relevant when assessing the performance of a particular design carries a high cost, as the overall cost can be mitigated via the construction of a model to be queried in lieu of the available high-cost source. The construction of these models can sometimes employ other sources of information which are both cheaper and less accurate. The existence of these sources however poses the question of which sources should be used when constructing a model. Recent studies have attempted to characterise harmful data sources to guide practitioners in choosing when to ignore a certain source. These studies have done so in a synthetic setting, characterising sources using a large amount of data that is not available in practice. Some of these studies have also been shown to potentially suffer from bias in the benchmarks used in the analysis. In this study, we present a characterisation of harmful low-fidelity sources using only the limited data available to train a surrogate model. We employ recently developed benchmark filtering techniques to conduct a bias-free assessment, providing objectively varied benchmark suites of different sizes for future research. Analysing one of these benchmark suites with the technique known as Instance Space Analysis, we provide an intuitive visualisation of when a low-fidelity source should be used and use this analysis to provide guidelines that can be used in an applied industrial setting.