Neural Active Manifolds: Nonlinear Dimensionality Reduction for Uncertainty Quantification.

Area of Science:

• Computational Science
• Machine Learning
• Scientific Computing

Background:

• Computationally expensive mathematical models pose challenges for many scientific computing tasks.
• Existing dimensionality reduction techniques may not be suitable for complex, high-dimensional model outputs.

Purpose of the Study:

• To introduce a novel nonlinear dimensionality reduction technique for computationally expensive models.
• To develop a method that reduces model output variability using a neural active manifold (NeurAM).
• To enable efficient execution of outer loop many-query tasks in scientific computing.

Main Methods:

• Leveraging autoencoders to discover a one-dimensional neural active manifold (NeurAM).
• Simultaneously learning a surrogate model with inputs on the NeurAM.
• The method relies solely on model evaluations, not requiring gradient information.

Main Results:

• The NeurAM effectively captures model output variability.
• The framework facilitates multifidelity sampling estimators with reduced variance.
• Theoretical and numerical proofs demonstrate the efficacy of NeurAM for sampling strategies.

Conclusions:

• The proposed dimensionality reduction strategy offers significant advantages over existing methods.
• NeurAM provides an efficient approach for sensitivity analysis and uncertainty propagation.
• This method enhances the applicability of complex mathematical models in scientific research.