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Register a new userForecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations
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Deep learning has an increasing impact to assist research, allowing, for example, the discovery of novel materials. Until now, however, these artificial intelligence techniques have fallen short of discovering the full differential equation of an experimental physical system. Here we show that a dynamical neural network, trained on a minimal amount of data, can predict the behavior of spintronic devices with high accuracy and an extremely efficient simulation time, compared to the micromagnetic simulations that are usually employed to model them. For this purpose, we re-frame the formalism of Neural Ordinary Differential Equations (ODEs) to the constraints of spintronics: few measured outputs, multiple inputs and internal parameters. We demonstrate with Spin-Neural ODEs an acceleration factor over 200 compared to micromagnetic simulations for a complex problem -- the simulation of a reservoir computer made of magnetic skyrmions (20 minutes compared to three days). In a second realization, we show that we can predict the noisy response of experimental spintronic nano-oscillators to varying inputs after training Spin-Neural ODEs on five milliseconds of their measured response to different excitations. Spin-Neural ODE is a disruptive tool for developing spintronic applications in complement to micromagnetic simulations, which are time-consuming and cannot fit experiments when noise or imperfections are present. Spin-Neural ODE can also be generalized to other electronic devices involving dynamics.
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We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
Neural Ordinary Differential Equations (ODE) are a promising approach to learn dynamic models from time-series data in science and engineering applications. This work aims at learning Neural ODE for stiff systems, which are usually raised from chemical kinetic modeling in chemical and biological systems. We first show the challenges of learning neural ODE in the classical stiff ODE systems of Robertsons problem and propose techniques to mitigate the challenges associated with scale separations in stiff systems. We then present successful demonstrations in stiff systems of Robertsons problem and an air pollution problem. The demonstrations show that the usage of deep networks with rectified activations, proper scaling of the network outputs as well as loss functions, and stabilized gradient calculations are the key techniques enabling the learning of stiff neural ODE. The success of learning stiff neural ODE opens up possibilities of using neural ODEs in applications with widely varying time-scales, like chemical dynamics in energy conversion, environmental engineering, and the life sciences.
Understanding physical phenomena oftentimes means understanding the underlying dynamical system that governs observational measurements. While accurate prediction can be achieved with black box systems, they often lack interpretability and are less amenable for further expert investigation. Alternatively, the dynamics can be analysed via symbolic regression. In this paper, we extend the approach by (Udrescu et al., 2020) called AIFeynman to the dynamic setting to perform symbolic regression on ODE systems based on observations from the resulting trajectories. We compare this extension to state-of-the-art approaches for symbolic regression empirically on several dynamical systems for which the ground truth equations of increasing complexity are available. Although the proposed approach performs best on this benchmark, we observed difficulties of all the compared symbolic regression approaches on more complex systems, such as Cart-Pole.
We propose a continuous neural network architecture, termed Explainable Tensorized Neural Ordinary Differential Equations (ETN-ODE), for multi-step time series prediction at arbitrary time points. Unlike the existing approaches, which mainly handle univariate time series for multi-step prediction or multivariate time series for single-step prediction, ETN-ODE could model multivariate time series for arbitrary-step prediction. In addition, it enjoys a tandem attention, w.r.t. temporal attention and variable attention, being able to provide explainable insights into the data. Specifically, ETN-ODE combines an explainable Tensorized Gated Recurrent Unit (Tensorized GRU or TGRU) with Ordinary Differential Equations (ODE). The derivative of the latent states is parameterized with a neural network. This continuous-time ODE network enables a multi-step prediction at arbitrary time points. We quantitatively and qualitatively demonstrate the effectiveness and the interpretability of ETN-ODE on five different multi-step prediction tasks and one arbitrary-step prediction task. Extensive experiments show that ETN-ODE can lead to accurate predictions at arbitrary time points while attaining best performance against the baseline methods in standard multi-step time series prediction.
Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. Here, we explore the use of Neural Ordinary Differential Equations, a recently introduced family of continuous-depth, differentiable networks (Chen et al 2018), as a way to propagate latent-space dynamics in reduced order models. We compare their behavior with two classical non-intrusive methods based on proper orthogonal decomposition and radial basis function interpolation as well as dynamic mode decomposition. The test problems we consider include incompressible flow around a cylinder as well as real-world applications of shallow water hydrodynamics in riverine and estuarine systems. Our findings indicate that Neural ODEs provide an elegant framework for stable and accurate evolution of latent-space dynamics with a promising potential of extrapolatory predictions. However, in order to facilitate their widespread adoption for large-scale systems, significant effort needs to be directed at accelerating their training times. This will enable a more comprehensive exploration of the hyperparameter space for building generalizable Neural ODE approximations over a wide range of system dynamics.
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