Learning Generalizable Reconstruction of High-Dimensional Neural Dynamics

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

We introduce ZUNA1.1, a 380M-parameter diffusion autoencoder for flexible EEG signal reconstruction. ZUNA1.1 is capable of reconstructing variable length sequences of up to 30s, with an arbitrary number of EEG channels at arbitrary scalp locations, and can reconstruct arbitrary temporal intervals within channels in addition to reconstructing entire channels. We demonstrate that ZUNA1.1 performs at least on par with our earlier ZUNA1 model, while being far more flexible and capable of handling a wide range of reconstruction tasks. ZUNA1.1 continues to substantially outperform standard EEG denoising and reconstruction methods such as spherical spline interpolation, which is ubiquitously deployed in the MNE package. The ZUNA1.1 model is released open source under the permissive Apache 2.0 license.

Accelerated acquisition of fMRI enables enhanced detection of neurovascular (BOLD) activity in the brain, but image reconstruction becomes challenging with high k-space undersampling: Task-evoked BOLD signals are small in magnitude, which traditional anatomical MRI reconstruction methods fail to recover, as they favor spatial accuracy over temporal fidelity. We present DD-INR, a Dynamics-Driven Implicit Neural Representation framework tailored for accelerated fMRI that benefits from incoherent time-varying sampling and a tailored spatiotemporal prior, outperforming traditional methods, demonstrated in simulation and in-vivo acquisition, both in terms of image quality and retrieval of activation patterns. DD-INR achieves this by splitting the fMRI data into a static background and a temporally varying dynamic component, representing only the dynamics with a dedicated INR, thereby focusing the model's capacity on activation-relevant changes while remaining compact. In general, DD-INR provides a promising framework for accelerated fMRI reconstruction, with the potential to improve the sensitivity and robustness of fMRI studies within practical scan time limits. The source code is available at https://github.com/JoosenLi/DD-INR.