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De Rham cohomological architectures for PINNs and PDEs with topological holes
Andrew Gracyk
We attempt to alleviate failure modes of classical physics-informed neural networks (PINNs) in scenarios of topological pathologies corresponding to punctures, holes, poles, and vo…
Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold
Andrew Gracyk
We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization…
Shortcomings and capacities of real-constrained neural networks in complex spaces
Andrew Gracyk
We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. We use w…
Pseudo-differential-enhanced physics-informed neural networks
Andrew Gracyk
We present pseudo-differential enhanced physics-informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates…
Complex variational autoencoders admit Kähler structure
Andrew Gracyk
It has been discovered that latent-Euclidean variational autoencoders (VAEs) admit, in various capacities, Riemannian structure. We adapt these arguments but for complex VAEs with…
Inexact calculus of variations on the hyperspherical tangent bundle with connections to the attention mechanism
Andrew Gracyk
We offer a theoretical mathematical background through Lagrangian optimization on the unit hyperspherical manifold and its tangential structure. Our methods can be categorized as i…
Observability conditions for neural state-space models with eigenvalues and their roots of unity
Andrew Gracyk
We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba ar…
Variational autoencoders with latent high-dimensional steady geometric flows for dynamics
Andrew Gracyk
We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs wit…
Ricci flow regularization in latent spaces for the forward learning of partial differential equations
Andrew Gracyk
We present a manifold-based machine learning encoder-decoder method for learning dynamics in time, notably partial differential equations (PDEs), in which the manifold latent space…
GeONet: a neural operator for learning the Wasserstein geodesic
Andrew Gracyk, Xiaohui Chen
Optimal transport (OT) offers a versatile framework to compare complex data distributions in a geometrically meaningful way. Traditional methods for computing the Wasserstein dista…