On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamics

We investigate geometric regularization strategies for learned latent representations in encoder--decoder reduced-order models. In a fixed experimental setting for the advection--diffusion--reaction (ADR) equation, we model latent dynamics using a neural ODE and evaluate four regularization approaches applied during autoencoder pre-training: (a) near-isometry regularization of the decoder Jacobian, (b) a stochastic decoder gain penalty based on random directional gains, (c) a second-order directional curvature penalty, and (d) Stiefel projection of the first decoder layer. Across multiple seeds, we find that (a)--(c) often produce latent representations that make subsequent latent-dynamics training with a frozen autoencoder more difficult, especially for long-horizon rollouts, even when they improve local decoder smoothness or related sensitivity proxies. In contrast, (d) consistently improves conditioning-related diagnostics of the learned latent dynamics and tends to yield better rollout performance. We discuss the hypothesis that, in this setting, the downstream impact of latent-geometry mismatch outweighs the benefits of improved decoder smoothness.

Symmetry-Breaking Descent for Invariant Cost Functionals

We study the problem of reducing a task cost functional $W(S)$, defined over Sobolev-class signals $S$, when the cost is invariant under a global symmetry group $G \subset \mathrm{Diff}(M)$ and accessible only as a black-box. Such scenarios arise in machine learning, imaging, and inverse problems, where cost metrics reflect model outputs or performance scores but are non-differentiable and model-internal. We propose a variational method that exploits the symmetry structure to construct explicit, symmetry-breaking deformations of the input signal. A gauge field $\phi$, obtained by minimizing an auxiliary energy functional, induces a deformation $h = A_\phi[S]$ that generically lies transverse to the $G$-orbit of $S$. We prove that, under mild regularity, the cost $W$ strictly decreases along this direction -- either via Clarke subdifferential descent or by escaping locally flat plateaus. The exceptional set of degeneracies has zero Gaussian measure. Our approach requires no access to model gradients or labels and operates entirely at test time. It provides a principled tool for optimizing invariant cost functionals via Lie-algebraic variational flows, with applications to black-box models and symmetry-constrained tasks.

Dysarthria Normalization via Local Lie Group Transformations for Robust ASR

We present a geometry-driven method for normalizing dysarthric speech using local Lie group transformations of spectrograms. Time, frequency, and amplitude distortions are modeled as smooth, invertible deformations, parameterized by scalar fields and applied via exponential maps. A neural network is trained to infer these fields from synthetic distortions of typical speech-without using any pathological data. At test time, the model applies an approximate inverse to real dysarthric inputs. Despite zero-shot generalization, we observe substantial ASR gains, including up to 16 percentage points WER reduction on challenging TORGO samples, with no degradation on clean speech. This work introduces a principled, interpretable approach for robust speech recognition under motor speech disorders