OS-net: Orbitally Stable Neural Networks

We introduce OS-net (Orbitally Stable neural NETworks), a new family of neural network architectures specifically designed for periodic dynamical data. OS-net is a special case of Neural Ordinary Differential Equations (NODEs) and takes full advantage of the adjoint method based backpropagation method. Utilizing ODE theory, we derive conditions on the network weights to ensure stability of the resulting dynamics. We demonstrate the efficacy of our approach by applying OS-net to discover the dynamics underlying the R\"{o}ssler and Sprott's systems, two dynamical systems known for their period doubling attractors and chaotic behavior.

Targeted Adaptive Design

Modern advanced manufacturing and advanced materials design often require searches of relatively high-dimensional process control parameter spaces for settings that result in optimal structure, property, and performance parameters. The mapping from the former to the latter must be determined from noisy experiments or from expensive simulations. We abstract this problem to a mathematical framework in which an unknown function from a control space to a design space must be ascertained by means of expensive noisy measurements, which locate optimal control settings generating desired design features within specified tolerances, with quantified uncertainty. We describe targeted adaptive design (TAD), a new algorithm that performs this optimal sampling task. TAD creates a Gaussian process surrogate model of the unknown mapping at each iterative stage, proposing a new batch of control settings to sample experimentally and optimizing the updated log-predictive likelihood of the target design. TAD either stops upon locating a solution with uncertainties that fit inside the tolerance box or uses a measure of expected future information to determine that the search space has been exhausted with no solution. TAD thus embodies the exploration-exploitation tension in a manner that recalls, but is essentially different from, Bayesian optimization and optimal experimental design.

Approximating periodic functions and solving differential equations using a novel type of Fourier Neural Networks

Recently, machine learning tools in particular neural networks have been widely used to solve differential equations. One main advantage of using machine learning, in this case, is that one does not need to mesh the computational domain and can instead randomly draw data points to solve the differential equations of interest. In this work, we propose a simple neural network to approximate low-frequency periodic functions or seek such solutions of differential equations. To this end, we build a Fourier Neural Network (FNN) represented as a shallow neural network (i.e with one hidden layer) based on the Fourier Decomposition. As opposed to traditional neural networks, which feature activation functions such as the sigmoid, logistic, ReLU, hyperbolic tangent and softmax functions, Fourier Neural Networks are composed using sinusoidal activation functions. We propose a strategy to initialize the weights of this FNN and showcase its performance against traditional networks for function approximations and differential equations solutions.