Neural active manifolds: nonlinear dimensionality reduction for uncertainty quantification

TL;DR

This work addresses the challenge of uncertainty quantification for computationally expensive, high-dimensional models by introducing Neural Active Manifolds (NeurAM), a nonlinear dimensionality reduction framework based on autoencoders augmented with a latent-space surrogate. The method learns a one-dimensional manifold that captures the full variability of the model output without requiring model gradients, enabling efficient sensitivity analysis and multifidelity uncertainty propagation. Key contributions include a jointly trained encoder–decoder–surrogate architecture, local and global sensitivity indices along the learned manifold, and a nonlinear MFMC approach that reparameterizes the low-fidelity model to boost cross-fidelity correlation, with a theoretical result showing superior correlation under idealized conditions. Numerical experiments on simple 2D problems, the Hartmann problem, and a cardiac electrophysiology model demonstrate that NeurAM often outperforms active subspaces and active manifolds, provides robust surrogate accuracy, and yields variance reductions in multifidelity estimators, highlighting its potential for accelerating many-query uncertainty quantification tasks.

Abstract

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, through the aid of a simultaneously learnt surrogate model with inputs on this manifold. Our method only relies on model evaluations and does not require the knowledge of gradients. The proposed dimensionality reduction framework can then be applied to assist outer loop many-query tasks in scientific computing, like sensitivity analysis and multifidelity uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy and highlight its advantages with respect to existing approaches in the literature.

Figures (17)

• Figure 1: Schematic representation of the NeurAM architecture, where the symbols $\simeq$ are used to indicate the three terms in the loss function in equation \ref{['eq:loss']}.
• Figure 2: Schematic representation of the multifidelity uncertainty propagation pipeline based on NeurAM.
• Figure 3: Analytical results for the two-dimensional model $Q$. Left: projection of input samples on the one-dimensional NeurAM. Right: parity plot between the exact and surrogate models.
• Figure 4: Approximation errors (MAE and MSE) in equation \ref{['eq:errors_def']} as functions of the size of the training set $N$ for the two-dimensional model $\mathcal{Q}$. Left: error $e_1$ due to the dimensionality reduction. Right: error $e_2$ of the surrogate model.
• Figure 5: Results for the two-dimensional models $\mathcal{Q}_1, \mathcal{Q}_2, \mathcal{Q}_3$, for one realization of the NeurAM. Top: projection of input samples on the one-dimensional NeurAM. Bottom: parity plot between the exact and surrogate models.

Theorems & Definitions (12)

• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 3.1
• Remark 3.2
• Theorem 3.3