  High-dimensional spatio-temporal dynamics can often be encoded in a
low-dimensional subspace. Engineering applications for modeling,
characterization, design, and control of such large-scale systems often rely on
dimensionality reduction to make solutions computationally tractable in
real-time. Common existing paradigms for dimensionality reduction include
linear methods, such as the singular value decomposition (SVD), and nonlinear
methods, such as variants of convolutional autoencoders (CAE). However, these
encoding techniques lack the ability to efficiently represent the complexity
associated with spatio-temporal data, which often requires variable geometry,
non-uniform grid resolution, adaptive meshing, and/or parametric dependencies.
To resolve these practical engineering challenges, we propose a general
framework called Neural Implicit Flow (NIF) that enables a mesh-agnostic,
low-rank representation of large-scale, parametric, spatial-temporal data. NIF
consists of two modified multilayer perceptrons (MLPs): (i) ShapeNet, which
isolates and represents the spatial complexity, and (ii) ParameterNet, which
accounts for any other input complexity, including parametric dependencies,
time, and sensor measurements. We demonstrate the utility of NIF for parametric
surrogate modeling, enabling the interpretable representation and compression
of complex spatio-temporal dynamics, efficient many-spatial-query tasks, and
improved generalization performance for sparse reconstruction.
