arXiv subjects

Sébastien da Veiga

Publications and source records attributed to Sébastien da Veiga.

11 recordsLinked to original sources

arXiv2026

Multi-output Gaussian process prediction of physical fields under linear equality constraints

We address the simultaneous prediction of multiple high-dimensional physical fields governed by linear equality constraints, a setting that arises in many real-world applications in physics machine learning. Gaussian process (GP) regression is a widely used surrogate modeling approach due to its effectiveness in small-sample regimes and its ability to provide uncertainty quantification. However, applying GP models in this setting raises two major challenges: the high dimensionality of the discretized output fields and the enforcement of the physical constraint in predictions. For the latter, a common strategy consists in deducing one output from the others via the constraint relation. Through a benchmark, we show that this deductive approach is sensitive to the arbitrary choice of which output to deduce, affecting both predictive accuracy and uncertainty quantification. Consequently, there is a need for an approach that treats all fields symmetrically while strictly respecting the underlying physics. Motivated by these limitations, we propose a robust framework for jointly modeling constrained multi-field data. Our approach first leverages a specific PCA procedure for multi-field data, coined row-wise PCA, which has the interesting property of preserving the constraint in the latent space. Since standard PCA strategies for multi-field data do not preserve such constraints, we investigate theoretically the optimality of the row-wise choice. In a second step, we consider a linearly-constrained multi-output GP approach based on a specific kernel parametrization which is trained on the latent space of row-wise PCA. The proposed framework is validated on a population dynamics problem and on an industrial CFD application, which involves the prediction of Reynolds stress tensor components under the incompressibility constraint.

stat.ML↗

arXiv2025

Scalable and adaptive prediction bands with kernel sum-of-squares

Conformal Prediction (CP) is a popular framework for constructing prediction bands with valid coverage in finite samples, while being free of any distributional assumption. A well-known limitation of conformal prediction is the lack of adaptivity, although several works introduced practically efficient alternate procedures. In this work, we build upon recent ideas that rely on recasting the CP problem as a statistical learning problem, directly targeting coverage and adaptivity. This statistical learning problem is based on reproducible kernel Hilbert spaces (RKHS) and kernel sum-of-squares (SoS) methods. First, we extend previous results with a general representer theorem and exhibit the dual formulation of the learning problem. Crucially, such dual formulation can be solved efficiently by accelerated gradient methods with several hundreds or thousands of samples, unlike previous strategies based on off-the-shelf semidefinite programming algorithms. Second, we introduce a new hyperparameter tuning strategy tailored specifically to target adaptivity through bounds on test-conditional coverage. This strategy, based on the Hilbert-Schmidt Independence Criterion (HSIC), is introduced here to tune kernel lengthscales in our framework, but has broader applicability since it could be used in any CP algorithm where the score function is learned. Finally, extensive experiments are conducted to show how our method compares to related work. All figures can be reproduced with the accompanying code.

cs.LG↗

arXiv2025

Learning signals defined on graphs with optimal transport and Gaussian process regression

In computational physics, machine learning has now emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. Outputs in such supervised problems are signals defined on meshes, and a natural question is the extension of general scalar output regression models to such complex outputs. Changes between input geometries in terms of both size and adjacency structure in particular make this transition non-trivial. In this work, we propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. The methodology relies on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty quantification and active learning. Numerical experiments highlight the efficiency of the method to solve real problems in fluid dynamics and solid mechanics.

stat.ML↗

arXiv2024

Gaussian process regression with Sliced Wasserstein Weisfeiler-Lehman graph kernels

Supervised learning has recently garnered significant attention in the field of computational physics due to its ability to effectively extract complex patterns for tasks like solving partial differential equations, or predicting material properties. Traditionally, such datasets consist of inputs given as meshes with a large number of nodes representing the problem geometry (seen as graphs), and corresponding outputs obtained with a numerical solver. This means the supervised learning model must be able to handle large and sparse graphs with continuous node attributes. In this work, we focus on Gaussian process regression, for which we introduce the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel. In contrast to existing graph kernels, the proposed SWWL kernel enjoys positive definiteness and a drastic complexity reduction, which makes it possible to process datasets that were previously impossible to handle. The new kernel is first validated on graph classification for molecular datasets, where the input graphs have a few tens of nodes. The efficiency of the SWWL kernel is then illustrated on graph regression in computational fluid dynamics and solid mechanics, where the input graphs are made up of tens of thousands of nodes.

stat.ML↗

arXiv2022

MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA

Variable importance measures are the main tools to analyze the black-box mechanisms of random forests. Although the mean decrease accuracy (MDA) is widely accepted as the most efficient variable importance measure for random forests, little is known about its statistical properties. In fact, the definition of MDA varies across the main random forest software. In this article, our objective is to rigorously analyze the behavior of the main MDA implementations. Consequently, we mathematically formalize the various implemented MDA algorithms, and then establish their limits when the sample size increases. This asymptotic analysis reveals that these MDA versions differ as importance measures, since they converge towards different quantities. More importantly, we break down these limits into three components: the first two terms are related to Sobol indices, which are well-defined measures of a covariate contribution to the response variance, widely used in the sensitivity analysis field, as opposed to the third term, whose value increases with dependence within covariates. Thus, we theoretically demonstrate that the MDA does not target the right quantity to detect influential covariates in a dependent setting, a fact that has already been noticed experimentally. To address this issue, we define a new importance measure for random forests, the Sobol-MDA, which fixes the flaws of the original MDA, and consistently estimates the accuracy decrease of the forest retrained without a given covariate, but with an efficient computational cost. The Sobol-MDA empirically outperforms its competitors on both simulated and real data for variable selection. An open source implementation in R and C++ is available online.

stat.ML↗

arXiv2022

SHAFF: Fast and consistent SHApley eFfect estimates via random Forests

Interpretability of learning algorithms is crucial for applications involving critical decisions, and variable importance is one of the main interpretation tools. Shapley effects are now widely used to interpret both tree ensembles and neural networks, as they can efficiently handle dependence and interactions in the data, as opposed to most other variable importance measures. However, estimating Shapley effects is a challenging task, because of the computational complexity and the conditional expectation estimates. Accordingly, existing Shapley algorithms have flaws: a costly running time, or a bias when input variables are dependent. Therefore, we introduce SHAFF, SHApley eFfects via random Forests, a fast and accurate Shapley effect estimate, even when input variables are dependent. We show SHAFF efficiency through both a theoretical analysis of its consistency, and the practical performance improvements over competitors with extensive experiments. An implementation of SHAFF in C++ and R is available online.

stat.ML↗

arXiv2021

Interpretable Random Forests via Rule Extraction

We introduce SIRUS (Stable and Interpretable RUle Set) for regression, a stable rule learning algorithm which takes the form of a short and simple list of rules. State-of-the-art learning algorithms are often referred to as "black boxes" because of the high number of operations involved in their prediction process. Despite their powerful predictivity, this lack of interpretability may be highly restrictive for applications with critical decisions at stake. On the other hand, algorithms with a simple structure-typically decision trees, rule algorithms, or sparse linear models-are well known for their instability. This undesirable feature makes the conclusions of the data analysis unreliable and turns out to be a strong operational limitation. This motivates the design of SIRUS, which combines a simple structure with a remarkable stable behavior when data is perturbed. The algorithm is based on random forests, the predictive accuracy of which is preserved. We demonstrate the efficiency of the method both empirically (through experiments) and theoretically (with the proof of its asymptotic stability). Our R/C++ software implementation sirus is available from CRAN.

stat.ML↗

arXiv2021

Kernel-based ANOVA decomposition and Shapley effects -- Application to global sensitivity analysis

Global sensitivity analysis is the main quantitative technique for identifying the most influential input variables in a numerical simulation model. In particular when the inputs are independent, Sobol' sensitivity indices attribute a portion of the output of interest variance to each input and all possible interactions in the model, thanks to a functional ANOVA decomposition. On the other hand, moment-independent sensitivity indices focus on the impact of input variables on the whole output distribution instead of the variance only, thus providing complementary insight on the inputs / output relationship. Unfortunately they do not enjoy the nice decomposition property of Sobol' indices and are consequently harder to analyze. In this paper, we introduce two moment-independent indices based on kernel-embeddings of probability distributions and show that the RKHS framework used for their definition makes it possible to exhibit a kernel-based ANOVA decomposition. This is the first time such a desirable property is proved for sensitivity indices apart from Sobol' ones. When the inputs are dependent, we also use these new sensitivity indices as building blocks to design kernel-embedding Shapley effects which generalize the traditional variance-based ones used in sensitivity analysis. Several estimation procedures are discussed and illustrated on test cases with various output types such as categorical variables and probability distributions. All these examples show their potential for enhancing traditional sensitivity analysis with a kernel point of view.

math.ST↗

arXiv2020

SIRUS: Stable and Interpretable RUle Set for Classification

State-of-the-art learning algorithms, such as random forests or neural networks, are often qualified as "black-boxes" because of the high number and complexity of operations involved in their prediction mechanism. This lack of interpretability is a strong limitation for applications involving critical decisions, typically the analysis of production processes in the manufacturing industry. In such critical contexts, models have to be interpretable, i.e., simple, stable, and predictive. To address this issue, we design SIRUS (Stable and Interpretable RUle Set), a new classification algorithm based on random forests, which takes the form of a short list of rules. While simple models are usually unstable with respect to data perturbation, SIRUS achieves a remarkable stability improvement over cutting-edge methods. Furthermore, SIRUS inherits a predictive accuracy close to random forests, combined with the simplicity of decision trees. These properties are assessed both from a theoretical and empirical point of view, through extensive numerical experiments based on our R/C++ software implementation sirus available from CRAN.

stat.ML↗

arXiv2020

Towards new cross-validation-based estimators for Gaussian process regression: efficient adjoint computation of gradients

We consider the problem of estimating the parameters of the covariance function of a Gaussian process by cross-validation. We suggest using new cross-validation criteria derived from the literature of scoring rules. We also provide an efficient method for computing the gradient of a cross-validation criterion. To the best of our knowledge, our method is more efficient than what has been proposed in the literature so far. It makes it possible to lower the complexity of jointly evaluating leave-one-out criteria and their gradients.

stat.CO↗

arXiv2019

Sequential model aggregation for production forecasting

Production forecasting is a key step to design the future development of a reservoir. A classical way to generate such forecasts consists in simulating future production for numerical models representative of the reservoir. However, identifying such models can be very challenging as they need to be constrained to all available data. In particular, they should reproduce past production data, which requires to solve a complex non-linear inverse problem. In this paper, we thus propose to investigate the potential of machine learning algorithms to predict the future production of a reservoir based on past production data without model calibration. We focus more specifically on robust online aggregation, a deterministic approach that provides a robust framework to make forecasts on a regular basis. This method does not rely on any specific assumption or need for stochastic modeling. Forecasts are first simulated for a set of base reservoir models representing the prior uncertainty, and then combined to predict production at the next time step. The weight associated to each forecast is related to its past performance. Three different algorithms are considered for weight computations: the exponentially weighted average algorithm, ridge regression and the Lasso regression. They are applied on a synthetic reservoir case study, the Brugge case, for sequential predictions. To estimate the potential of development scenarios, production forecasts are needed on long periods of time without intermediary data acquisition. An extension of the deterministic aggregation approach is thus proposed in this paper to provide such multi-step-ahead forecasts.

stat.ML↗