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PhILMs: Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems
The landscape of computational science and engineering is continually evolving, with the challenge of high-dimensional regression problems standing as a significant hurdle in numerous scientific endeavors. Addressing this challenge, our research, funded by this award, has led to the development of an innovative computational framework known as Probabilistic Partition of Unity Networks (PPOU-Nets). This initiative represents a collaborative effort to harness the potential of mathematics and physics-informed machine learning in tackling multiscale and multiphysics problems prevalent in high-dimensional spaces. Through this work, we have proposed a novel methodology that seamlessly integrates adaptive dimensionality reduction and a mixture of experts model, thereby facilitating a more efficient and accurate approximation of complex functions. This research effort has not only advanced the state of computational science but also opened new avenues for exploration in quantum computing and beyond. This report outlines the motivation, methodology, key findings, and implications of our work, underscoring our contributions to the broader scientific community and the potential pathways for future research.
Data-driven Whitney forms for structure-preserving control volume analysis
Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.
PCAfold 2.0—Novel tools and algorithms for low-dimensional manifold assessment and optimization
We describe an update to our open-source Python package, PCAfold, designed to help researchers generate, analyze and improve low-dimensional data manifolds. In the current version, PCAfold 2.0, we introduce novel tools and algorithms for assessing and optimizing low-dimensional manifolds. This includes a method that generates a “map” of local feature sizes that can help pinpoint researchers to problematic regions on a manifold. We introduce a novel cost function that characterizes the quality of a manifold topology with a single number. We develop two algorithms for feature selection based on principal component analysis (PCA) that use the cost function as an objective function to minimize. We introduce a quantity of interest (QoI)-aware dimensionality reduction strategy where data projections are computed using an artificial neural network and are directly optimized towards representing various projection-independent and projection-dependent QoIs. We also introduce an implementation of partition of unity networks (POUnets) for efficient reconstruction of QoIs from low-dimensional manifolds based on combining neural network classification with localized polynomial regression. Our software can be broadly applicable in all domains of science and engineering that aim to reduce data dimensionality, as well as in the fundamental research on representation learning.
Global Simulations of Phase State and Equilibration Time Scales of Secondary Organic Aerosols with GEOS-Chem
The phase state of secondary organic aerosols (SOA) can range from liquid through amorphous semisolid to glassy solid, which is important to consider as it influences various multiphase processes including SOA formation and partitioning, multiphase chemistry, and cloud activation. In this study, we simulate the glass transition temperature and viscosity of SOA over the globe using the global chemical transport model, GEOS-Chem. The simulated spatial distributions show that SOA at the surface exist as liquid over equatorial regions and oceans, semisolid in the midlatitude continental regions, and glassy solid over lands with low relative humidity. The predicted SOA viscosities are mostly consistent with the available measurements. In the free troposphere, SOA particles are mostly predicted to be semisolid at 850 hPa and glassy solid at 500 hPa, except over tropical regions including Amazonia, where SOA are predicted to be low viscous. Phase state also exhibits seasonal variation with a higher frequency of semisolid and solid particles in winter compared to warmer seasons. We calculate equilibration time scales of SOA partitioning (τ eq ) and effective mass accommodation coefficient (α eff ), indicating that τ eq is shorter than the chemical time step of GEOS-Chem of 20 min and αeff is close to unity for most locations at the surface level, supporting the application of equilibrium SOA partitioning. However, τ eq is prolonged and α eff is lowered over drylands and most regions in the upper troposphere, suggesting that kinetically limited growth would need to be considered for these regions in future large-scale model studies.