Learning Hamiltonian Dynamics with Bayesian Data Assimilation

In this paper, we develop a neural network-based approach for time-series prediction in unknown Hamiltonian dynamical systems. Our approach leverages a surrogate model and learns the system dynamics using generalized coordinates (positions) and their conjugate momenta while preserving a constant Hamiltonian. To further enhance long-term prediction accuracy, we introduce an Autoregressive Hamiltonian Neural Network, which incorporates autoregressive prediction errors into the training objective. Additionally, we employ Bayesian data assimilation to refine predictions in real-time using online measurement data. Numerical experiments on a spring-mass system and highly elliptic orbits under gravitational perturbations demonstrate the effectiveness of the proposed method, highlighting its potential for accurate and robust long-term predictions.

Neural Hamilton: Can A.I. Understand Hamiltonian Mechanics?

We propose a novel framework based on neural network that reformulates classical mechanics as an operator learning problem. A machine directly maps a potential function to its corresponding trajectory in phase space without solving the Hamilton equations. Most notably, while conventional methods tend to accumulate errors over time through iterative time integration, our approach prevents error propagation. Two newly developed neural network architectures, namely VaRONet and MambONet, are introduced to adapt the Variational LSTM sequence-to-sequence model and leverage the Mamba model for efficient temporal dynamics processing. We tested our approach with various 1D physics problems: harmonic oscillation, double-well potentials, Morse potential, and other potential models outside the training data. Compared to traditional numerical methods based on the fourth-order Runge-Kutta (RK4) algorithm, our model demonstrates improved computational efficiency and accuracy. Code is available at: https://github.com/Axect/Neural_Hamilton

HyperbolicLR: Epoch insensitive learning rate scheduler

This study proposes two novel learning rate schedulers: the Hyperbolic Learning Rate Scheduler (HyperbolicLR) and the Exponential Hyperbolic Learning Rate Scheduler (ExpHyperbolicLR). These schedulers attempt to address the inconsistent learning curves often observed in conventional schedulers when adjusting the number of epochs. By leveraging the asymptotic behavior of hyperbolic curves, the proposed schedulers maintain more consistent learning curves across varying epoch settings. The HyperbolicLR algorithm directly applies this property to the epoch-learning rate space, while the ExpHyperbolicLR maps this concept onto the exponential space of epochs and learning rates. To evaluate the performance of these schedulers, first we found the optimal hyperparameters for each scheduler on a small number of epochs, fixed these values, and compared their performance as the number of epochs increased. Our experimental results on various deep learning tasks and architectures demonstrate that both HyperbolicLR and ExpHyperbolicLR maintain more consistent performance improvements compared to conventional schedulers as the number of epochs increases. These findings suggest that our hyperbolic-based learning rate schedulers offer a more robust and efficient approach to training deep neural networks, especially in scenarios where computational resources or time constraints limit extensive hyperparameter searches.

Unsupervised sequence-to-sequence learning for automatic signal quality assessment in multi-channel electrical impedance-based hemodynamic monitoring

This study proposes an unsupervised sequence-to-sequence learning approach that automatically assesses the motion-induced reliability degradation of the cardiac volume signal (CVS) in multi-channel electrical impedance-based hemodynamic monitoring. The proposed method attempts to tackle shortcomings in existing learning-based assessment approaches, such as the requirement of manual annotation for motion influence and the lack of explicit mechanisms for realizing motion-induced abnormalities under contextual variations in CVS over time. By utilizing long-short term memory and variational auto-encoder structures, an encoder--decoder model is trained not only to self-reproduce an input sequence of the CVS but also to extrapolate the future in a parallel fashion. By doing so, the model can capture contextual knowledge lying in a temporal CVS sequence while being regularized to explore a general relationship over the entire time-series. A motion-influenced CVS of low-quality is detected, based on the residual between the input sequence and its neural representation with a cut--off value determined from the two-sigma rule of thumb over the training set. Our experimental observations validated two claims: (i) in the learning environment of label-absence, assessment performance is achievable at a competitive level to the supervised setting, and (ii) the contextual information across a time series of CVS is advantageous for effectively realizing motion-induced unrealistic distortions in signal amplitude and morphology. We also investigated the capability as a pseudo-labeling tool to minimize human-craft annotation by preemptively providing strong candidates for motion-induced anomalies. Empirical evidence has shown that machine-guided annotation can reduce inevitable human-errors during manual assessment while minimizing cumbersome and time-consuming processes.