Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

Soize, Christian; Ghanem, R.

doi:10.1016/j.cma.2021.113777

Home/Archive/DOE OSTI

DOE OSTI · 3024508

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

Soize, Christian·Ghanem, R.

View original source ↗Read source document ↗Export metadata ↓

Abstract

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. Here, the last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Explore related subjects

Machine learning PLoM Partial differential equations Probabilistic learning Small dataset Uncertainty quantification

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Soize, Christian, Ghanem, R.. 2021-03-24. Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets. https://doi.org/10.1016/j.cma.2021.113777

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related reports

DOE OSTI2026

Predicting Adaptively Chosen Observables in Quantum Systems

Recent advances have demonstrated that 𝒪⁡(log 𝑀) measurements suffice to predict 𝑀 properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that Ω⁡(√𝑀) samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of 𝑀 adaptively chosen local and Pauli observables, where the system size scales exponentially and polynomially in 𝑀, respectively. We also present computationally efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only 𝒪⁡(log 𝑀) samples, independent of system size. These results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide algorithmic tools to safeguard against erroneous predictions in quantum experiments.

Machine learning↗

DOE OSTI2025

Using active learning to improve quasar identification for the DESI spectra processing pipeline

The Dark Energy Spectroscopic Instrument (DESI) survey uses an automatic spectral classification pipeline to classify spectra. QuasarNET is a convolutional neural network used as part of this pipeline originally trained using data from the Baryon Oscillation Spectroscopic Survey (BOSS). In this paper we implement an active learning algorithm to optimally select spectra to use for training a new version of the QuasarNET weights file using only DESI data, with the goal of improving classification accuracy. This active learning algorithm includes a novel outlier rejection step using a Self-Organizing Map to ensure we label spectra representative of the larger quasar sample observed in DESI. We perform two iterations of the active learning pipeline, assembling a final dataset of 5600 labeled spectra, a small subset of the approximately 1.3 million quasar targets in DESI's Data Release 1. When splitting the spectra into training and validation subsets we achieve similar performance to the previously trained weights file in completeness and purity calculated on the validation dataset but do so with less than one tenth of the amount of training data. The new weights also more consistently classify objects in the same way when used on unlabeled data compared to the old weights file. In the process of improving QuasarNET's classification accuracy we discovered a systemic error in QuasarNET's redshift estimation and used our findings to improve our understanding of QuasarNET's redshifts.

Machine learning↗

DOE OSTI2025

Learning nuclear cross sections across the chart of nuclides with graph neural networks

We explore the use of deep learning techniques to learn how nuclear cross sections change as we add or remove protons and neutrons. As a proof of principle, we focus on the neutron-induced reactions in the fast energy regime. Our approach follows a two-stage learning framework. First, we apply representation learning to encode cross section data into a latent space using either variational autoencoders (VAEs) or implicit neural representations (INRs). Then, we train graph neural networks (GNNs) on the resulting embeddings to predict missing values across the nuclear chart by leveraging the topological structure of neighboring isotopes. We demonstrate accurate cross section predictions within a 9 × 9 block of missing nuclei. We also find that the optimal GNN training strategy depends on the type of latent representation used, with VAE embeddings performing best under end-to-end optimization in the original space, while INR embeddings achieve better results when the GNN is trained only in the latent space. Furthermore, using clustering algorithms, we map groups of latent vectors into regions of the nuclear chart and show that VAEs and INRs can discover some of the neutron magic numbers. These findings suggest that deep-learning models based on the representation encoding of cross sections combined with graph neural networks hold significant potential in augmenting nuclear theory models, e.g., by providing reliable estimates of covariances of cross sections, including cross-material covariances.

Machine learning↗