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Graph neural networks (GNNs) are deep convolutional architectures consisting of layers composed by graph convolutions and pointwise nonlinearities. Due to their invariance and stability properties, GNNs are provably successful at learning representations from network data. However, training them requires matrix computations which can be expensive for large graphs. To address this limitation, we investigate the ability of GNNs to be transferred across graphs. We consider graphons, which are both graph limits and generative models for weighted and stochastic graphs, to define limit objects of graph convolutions and GNNs -- graphon convolutions and graphon neural networks (WNNs) -- which we use as generative models for graph convolutions and GNNs. We show that these graphon filters and WNNs can be approximated by graph filters and GNNs sampled from them on weighted and stochastic graphs. Using these results, we then derive error bounds for transferring graph filters and GNNs across such graphs. These bounds show that transferability increases with the graph size, and reveal a tradeoff between transferability and spectral discriminability which in GNNs is alleviated by the pointwise nonlinearities. These findings are further verified empirically in numerical experiments in movie recommendation and decentralized robot control.

In many areas, such as the physical sciences, life sciences, and finance, control approaches are used to achieve a desired goal in complex dynamical systems governed by differential equations. In this work we formulate the problem of controlling stochastic partial differential equations (SPDE) as a reinforcement learning problem. We present a learning-based, distributed control approach for online control of a system of SPDEs with high dimensional state-action space using deep deterministic policy gradient method. We tested the performance of our method on the problem of controlling the stochastic Burgers' equation, describing a turbulent fluid flow in an infinitely large domain.

On-policy deep reinforcement learning algorithms have low data utilization and require significant experience for policy improvement. This paper proposes a proximal policy optimization algorithm with prioritized trajectory replay (PTR-PPO) that combines on-policy and off-policy methods to improve sampling efficiency by prioritizing the replay of trajectories generated by old policies. We first design three trajectory priorities based on the characteristics of trajectories: the first two being max and mean trajectory priorities based on one-step empirical generalized advantage estimation (GAE) values and the last being reward trajectory priorities based on normalized undiscounted cumulative reward. Then, we incorporate the prioritized trajectory replay into the PPO algorithm, propose a truncated importance weight method to overcome the high variance caused by large importance weights under multistep experience, and design a policy improvement loss function for PPO under off-policy conditions. We evaluate the performance of PTR-PPO in a set of Atari discrete control tasks, achieving state-of-the-art performance. In addition, by analyzing the heatmap of priority changes at various locations in the priority memory during training, we find that memory size and rollout length can have a significant impact on the distribution of trajectory priorities and, hence, on the performance of the algorithm.

Traffic forecasting is one of the most popular spatio-temporal tasks in the field of machine learning. A prevalent approach in the field is to combine graph convolutional networks and recurrent neural networks for the spatio-temporal processing. There has been fierce competition and many novel methods have been proposed. In this paper, we present the method of spatio-temporal graph neural controlled differential equation (STG-NCDE). Neural controlled differential equations (NCDEs) are a breakthrough concept for processing sequential data. We extend the concept and design two NCDEs: one for the temporal processing and the other for the spatial processing. After that, we combine them into a single framework. We conduct experiments with 6 benchmark datasets and 20 baselines. STG-NCDE shows the best accuracy in all cases, outperforming all those 20 baselines by non-trivial margins.

Molecular Dynamics (MD) simulation is a powerful tool for understanding the dynamics and structure of matter. Since the resolution of MD is atomic-scale, achieving long time-scale simulations with femtosecond integration is very expensive. In each MD step, numerous redundant computations are performed which can be learnt and avoided. These redundant computations can be surrogated and modeled by a deep learning model like a Graph Neural Network (GNN). In this work, we developed a GNN Accelerated Molecular Dynamics (GAMD) model that achieves fast and accurate force predictions and generates trajectories consistent with the classical MD simulations. Our results show that GAMD can accurately predict the dynamics of two typical molecular systems, Lennard-Jones (LJ) particles and Water (LJ+Electrostatics). GAMD's learning and inference are agnostic to the scale, where it can scale to much larger systems at test time. We also performed a comprehensive benchmark test comparing our implementation of GAMD to production-level MD softwares, where we showed GAMD is competitive with them on the large-scale simulation.

Understanding decentralized dynamics from collective behaviors in swarms is crucial for informing robot controller designs in artificial swarms and multiagent robotic systems. However, the complexity in agent-to-agent interactions and the decentralized nature of most swarms pose a significant challenge to the extraction of single-robot control laws from global behavior. In this work, we consider the important task of learning decentralized single-robot controllers based solely on the state observations of a swarm's trajectory. We present a general framework by adopting knowledge-based neural ordinary differential equations (KNODE) -- a hybrid machine learning method capable of combining artificial neural networks with known agent dynamics. Our approach distinguishes itself from most prior works in that we do not require action data for learning. We apply our framework to two different flocking swarms in 2D and 3D respectively, and demonstrate efficient training by leveraging the graphical structure of the swarms' information network. We further show that the learnt single-robot controllers can not only reproduce flocking behavior in the original swarm but also scale to swarms with more robots.

In this work, we consider the problem of deriving and incorporating accurate dynamic models for model predictive control (MPC) with an application to quadrotor control. MPC relies on precise dynamic models to achieve the desired closed-loop performance. However, the presence of uncertainties in complex systems and the environments they operate in poses a challenge in obtaining sufficiently accurate representations of the system dynamics. In this work, we make use of a deep learning tool, knowledge-based neural ordinary differential equations (KNODE), to augment a model obtained from first principles. The resulting hybrid model encompasses both a nominal first-principle model and a neural network learnt from simulated or real-world experimental data. Using a quadrotor, we benchmark our hybrid model against a state-of-the-art Gaussian Process (GP) model and show that the hybrid model provides more accurate predictions of the quadrotor dynamics and is able to generalize beyond the training data. To improve closed-loop performance, the hybrid model is integrated into a novel MPC framework, known as KNODE-MPC. Results show that the integrated framework achieves 60.2% improvement in simulations and more than 21% in physical experiments, in terms of trajectory tracking performance.

Controllable generation is one of the key requirements for successful adoption of deep generative models in real-world applications, but it still remains as a great challenge. In particular, the compositional ability to generate novel concept combinations is out of reach for most current models. In this work, we use energy-based models (EBMs) to handle compositional generation over a set of attributes. To make them scalable to high-resolution image generation, we introduce an EBM in the latent space of a pre-trained generative model such as StyleGAN. We propose a novel EBM formulation representing the joint distribution of data and attributes together, and we show how sampling from it is formulated as solving an ordinary differential equation (ODE). Given a pre-trained generator, all we need for controllable generation is to train an attribute classifier. Sampling with ODEs is done efficiently in the latent space and is robust to hyperparameters. Thus, our method is simple, fast to train, and efficient to sample. Experimental results show that our method outperforms the state-of-the-art in both conditional sampling and sequential editing. In compositional generation, our method excels at zero-shot generation of unseen attribute combinations. Also, by composing energy functions with logical operators, this work is the first to achieve such compositionality in generating photo-realistic images of resolution 1024x1024. Code is available at //github.com/NVlabs/LACE.

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

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.