The most widely used technology to identify the proteins present in a complex biological sample istandem mass spectrometry,which quickly produces alarge collection of spectra representative of thepeptides (i.e., protein subsequences) present in the original sample.

Identification and comparison of nonlinear dynamical systems using noisy and sparse experimental data is a vital task in many fields, however current methods are computationally expensive and prone to error due in part to the nonlinear nature of the likelihood surfaces induced. We present an accelerated sampling procedure which enables Bayesian inference of parameters in nonlinear ordinary and delay differential equations via the novel use of Gaussian processes (GP). Our method involves GP regression over time-series data, and the resulting derivative and time delay estimates make parameter inference possible without solving the dynamical system explicitly, resulting in dramatic savings of computational time. We demonstrate the speed and statistical accuracy of our approach using examples of both ordinary and delay differential equations, and provide a comprehensive comparison with current state of the art methods.

Identification and comparison of nonlinear dynamical systems using noisy and sparse experimental data is a vital task in many fields, however current methods are computationally expensive and prone to error due in part to the nonlinear nature of the likelihood surfaces induced. We present an accelerated sampling procedure which enables Bayesian inference of parameters in nonlinear ordinary and delay differential equations via the novel use of Gaussian processes (GP). Our method involves GP regression over time-series data, and the resulting derivative and time delay estimates make parameter inference possible without solving the dynamical system explicitly, resulting in dramatic savings of computational time. We demonstrate the speed and statistical accuracy of our approach using examples of both ordinary and delay differential equations, and provide a comprehensive comparison with current state of the art methods. Papers published at the Neural Information Processing Systems Conference.

In this work we systematically analyze general properties of differential equations used as machine learning models. We demonstrate that the gradient of the loss function with respect to to the hidden state can be considered as a generalized momentum conjugate to the hidden state, allowing application of the tools of classical mechanics. In addition, we show that not only residual networks, but also feedforward neural networks with small nonlinearities and the weights matrices deviating only slightly from identity matrices can be related to the differential equations. We propose a differential equation describing such networks and investigate its properties. 1 Introduction Deep learning is a form of machine learning that uses neural networks with many hidden layers [1, 2]. Deep learning models have dramatically improved speech recognition, visual object recognition, object detection and many other domains [2]. Since the number of layers in deep neural networks become large, it is possible to consider the layer number as a continuous variable [3] and represent the neural network by an differential equation. The connection between the neural networks and differential equations first appeared with an additive model for continuous time recurrent neural networks, described by the differential equations [4] τ i dx i dt x i n null j 1w j,iσ ( x j θ j) I i(t) . Hopfield's work [6] pioneered the analog computation of continuous time recurrent neural networks instead of digital computation using complex numerical algorithms on a digital computer. A Hopfield network has a quadratic form as an Lyapunov function for the activity dynamics. As a consequence, the state of the network evolves to a final state that is a minimum of the Lyapunov function when started in any initial state [7].

Physicists use differential equations to describe the physical dynamical world, and the solutions of these equations constitute our understanding of the world. During the hundreds of years, scientists developed several ways to solve these equations, i.e., the analytical solutions and the numerical solutions. However, for some complex equations, there may be no analytical solutions, and the numerical solutions may encounter the curse of the extreme computational cost if the accuracy is the first consideration. Solving equations is a high-level human intelligence work and a crucial step towards general artificial intelligence (AI), where deep reinforcement learning (DRL) may contribute. This work makes the first attempt of applying (DRL) to solve nonlinear differential equations both in discretized and continuous format with the governing equations (physical laws) embedded in the DRL network, including ordinary differential equations (ODEs) and partial differential equations (PDEs). The DRL network consists of an actor that outputs solution approximations policy and a critic that outputs the critic of the actor's output solution. Deterministic policy network is employed as the actor, and governing equations are embedded in the critic. The effectiveness of the DRL solver in Schr\"odinger equation, Navier-Stocks, Van der Pol equation, Burgers' equation and the equation of motion are discussed.

Many practical applications (e.g., plannning for cyber-physical systems) require reasoning about hybrid domains that contain both probabilistic and nondeterministic parametric uncertainty. In general, this is an undecidable problem. We use delta-satisfiability to sidestep undecidability, and we develop an algorithm that computes an enclosure for the range of probability of reaching a goal region in a given number of discrete steps. We utilize SMT techniques that enable reasoning in a safe way, i.e., the computed enclosure is formally guaranteed to contain the reachability probability. We demonstrate the usefulness of our technique on challenging nonlinear hybrid domains.

The question of how best to estimate a continuous probability density from finite data is an intriguing open problem at the interface of statistics and physics. Previous work has argued that this problem can be addressed in a natural way using methods from statistical field theory. Here I describe new results that allow this field-theoretic approach to be rapidly and deterministically computed in low dimensions, making it practical for use in day-to-day data analysis. Importantly, this approach does not impose a privileged length scale for smoothness of the inferred probability density, but rather learns a natural length scale from the data due to the tradeoff between goodness-of-fit and an Occam factor. Open source software implementing this method in one and two dimensions is provided.

Identification and comparison of nonlinear dynamical systems using noisy and sparse experimental data is a vital task in many fields, however current methods are computationally expensive and prone to error due in part to the nonlinear nature of the likelihood surfaces induced. We present an accelerated sampling procedure which enables Bayesian inference of parameters in nonlinear ordinary and delay differential equations via the novel use of Gaussian processes (GP). Our method involves GP regression over time-series data, and the resulting derivative and time delay estimates make parameter inference possible without solving the dynamical system explicitly, resulting in dramatic savings of computational time. We demonstrate the speed and statistical accuracy of our approach using examples of both ordinary and delay differential equations, and provide a comprehensive comparison with current state of the art methods.