Stochastic Runge-Kutta methods and adaptive SGD-G2 stochastic gradient descent

Probabilistic ODE Solvers with Runge-Kutta Means

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Ku…

Learning Hamiltonian Systems with Mono-Implicit Runge-Kutta Methods

Numerical integrators could be used to form interpolation conditions when training neural networks to approximate the vector field of an ordinary differential equation (ODE) from data. When numerical one-step schemes such as the Runge-Kutta methods are used to approximate the temporal discretization of an ODE with a known vector field, properties such as symmetry and stability are much studied. Here, we show that using mono-implicit Runge-Kutta methods of high order allows for accurate training of Hamiltonian neural networks on small datasets. This is demonstrated by numerical experiments wher…

Model-Agnostic Meta-Learning using Runge-Kutta Methods

Meta-learning has emerged as an important framework for learning new tasks from just a few examples. The success of any meta-learning model depends on (i) its fast adaptation to new tasks, as well as (ii) having a shared representation across similar tasks. Here we extend the model-agnostic meta-learning (MAML) framework introduced by Finn et al. (2017) to achieve improved performance by analyzing the temporal dynamics of the optimization procedure via the Runge-Kutta method. This method enables us to gain fine-grained control over the optimization and helps us achieve both the adaptation and …

Classification with Runge-Kutta networks and feature space augmentation

In this paper we combine an approach based on Runge-Kutta Nets considered in [Benning et al., J. Comput. Dynamics, 9, 2019] and a technique on augmenting the input space in [Dupont et al., NeurIPS, 2019] to obtain network architectures which show a better numerical performance for deep neural networks in point and image classification problems. The approach is illustrated with several examples implemented in PyTorch.

Convolutional Neural Networks combined with Runge-Kutta Methods

A convolutional neural network can be constructed using numerical methods for solving dynamical systems, since the forward pass of the network can be regarded as a trajectory of a dynamical system. However, existing models based on numerical solvers cannot avoid the iterations of implicit methods, which makes the models inefficient at inference time. In this paper, we reinterpret the pre-activation Residual Networks (ResNets) and their variants from the dynamical systems view. We consider that the iterations of implicit Runge-Kutta methods are fused into the training of these models. Moreover,…

基于 Runge-Kutta 数值方法的人脸识别新型稀疏 PCA 方法

Novel sparse PCA method via Runge Kutta numerical method(s) for face recognition

针对人脸识别问题，本文提出结合近端梯度法或Runge-Kutta数值方法实现稀疏PCA，并与分类算法相融合。实验表明该方法较标准PCA准确率更高，且基于Runge-Kutta的计算在速度上更具优势。