We present a method for learning generalized Hamiltonian decompositions of ordinary differential equations given a set of noisy time series measurements. Our method simultaneously learns a continuous time model and a scalar energy function for a general dynamical system. Learning predictive models in this form allows one to place strong, high-level, physics inspired priors onto the form of the learnt governing equations for general dynamical systems. Moreover, having shown how our method extends and unifies some previous work in deep learning with physics inspired priors, we present a novel method for learning continuous time models from the weak form of the governing equations which is less computationally taxing than standard adjoint methods.

This paper establishes a continuous time approximation, a piece-wise continuous differential equation, for the discrete Heavy-Ball (HB) momentum method with explicit discretization error. Investigating continuous differential equations has been a promising approach for studying the discrete optimization methods. Despite the crucial role of momentum in gradient-based optimization methods, the gap between the original discrete dynamics and the continuous time approximations due to the discretization error has not been comprehensively bridged yet. In this work, we study the HB momentum method in continuous time while putting more focus on the discretization error to provide additional theoretical tools to this area. In particular, we design a first-order piece-wise continuous differential equation, where we add a number of counter terms to account for the discretization error explicitly. As a result, we provide a continuous time model for the HB momentum method that allows the control of discretization error to arbitrary order of the step size. As an application, we leverage it to find a new implicit regularization of the directional smoothness and investigate the implicit bias of HB for diagonal linear networks, indicating how our results can be used in deep learning. Our theoretical findings are further supported by numerical experiments.

We present a method for learning generalized Hamiltonian decompositions of ordinary differential equations given a set of noisy time series measurements. Our method simultaneously learns a continuous time model and a scalar energy function for a general dynamical system. Learning predictive models in this form allows one to place strong, high-level, physics inspired priors onto the form of the learnt governing equations for general dynamical systems. Moreover, having shown how our method extends and unifies some previous work in deep learning with physics inspired priors, we present a novel method for learning continuous time models from the weak form of the governing equations which is less computationally taxing than standard adjoint methods.

Multiple sclerosis is a disease that affects the brain and spinal cord, it can lead to severe disability and has no known cure. The majority of prior work in machine learning for multiple sclerosis has been centered around using Magnetic Resonance Imaging scans or laboratory tests; these modalities are both expensive to acquire and can be unreliable. In a recent paper it was shown that disease progression can be predicted effectively using performance outcome measures and demographic data. In our work we build on this to investigate the modeling side, using continuous time models to predict progression. We benchmark four continuous time models using a publicly available multiple sclerosis dataset. We find that the best continuous model is often able to outperform the best benchmarked discrete time model. We also carry out an extensive ablation to discover the sources of performance gains, we find that standardizing existing features leads to a larger performance increase than interpolating missing features.

In this work, a Gaussian process regression(GPR) model incorporated with given physical information in partial differential equations(PDEs) is developed: physics-assisted Gaussian processes(PAGP). The targets of this model can be divided into two types of problem: finding solutions or discovering unknown coefficients of given PDEs with initial and boundary conditions. We introduce three different models: continuous time, discrete time and hybrid models. The given physical information is integrated into Gaussian process model through our designed GP loss functions. Three types of loss function are provided in this paper based on two different approaches to train the standard GP model. The first part of the paper introduces the continuous time model which treats temporal domain the same as spatial domain. The unknown coefficients in given PDEs can be jointly learned with GP hyper-parameters by minimizing the designed loss function. In the discrete time models, we first choose a time discretization scheme to discretize the temporal domain. Then the PAGP model is applied at each time step together with the scheme to approximate PDE solutions at given test points of final time. To discover unknown coefficients in this setting, observations at two specific time are needed and a mixed mean square error function is constructed to obtain the optimal coefficients. In the last part, a novel hybrid model combining the continuous and discrete time models is presented. It merges the flexibility of continuous time model and the accuracy of the discrete time model. The performance of choosing different models with different GP loss functions is also discussed. The effectiveness of the proposed PAGP methods is illustrated in our numerical section.

Finite mixture models have been used for unsupervised learning for some time, and their use within the semi-supervised paradigm is becoming more commonplace. Clickstream data is one of the various emerging data types that demands particular attention because there is a notable paucity of statistical learning approaches currently available. A mixture of first-order continuous time Markov models is introduced for unsupervised and semi-supervised learning of clickstream data. This approach assumes continuous time, which distinguishes it from existing mixture model-based approaches; practically, this allows account to be taken of the amount of time each user spends on each webpage. The approach is evaluated, and compared to the discrete time approach, using simulated and real data.