In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". 
In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.

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Universal Differential Equations for Scientific Machine Learning

Hamilton-Jacobi equations are frequently encountered in applications,e.g.,in control theory and differential games.Hamilton-Jacobi equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter.Similarity also exists for the multidimensional cases,and this is helpful in designing difference approximations.In this paper central weighted essentially non-oscillatory (CWENO) schemes for Hamilton-Jacobi equations are investigated,which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives.The schemes are numerically tested on a variety of one-dimensional problems,including a problem related to control optimization.High-order accuracy in smooth regions,high resolution of discontinuities in the derivatives,and convergence to viscosity solutions are observed.

High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations

Deep generative models (DGM) are neural networks with many hidden layers trained to approximate complicated, high-dimensional probability distributions using a large number of samples. When trained successfully, we can use the DGMs to estimate the likelihood of each observation and to create new samples from the underlying distribution. Developing DGMs has become one of the most hotly researched fields in artificial intelligence in recent years. The literature on DGMs has become vast and is growing rapidly. Some advances have even reached the public sphere, for example, the recent successes in generating realistic-looking images, voices, or movies; so-called deep fakes. Despite these successes, several mathematical and practical issues limit the broader use of DGMs: given a specific dataset, it remains challenging to design and train a DGM and even more challenging to find out why a particular model is or is not effective. To help advance the theoretical understanding of DGMs, we introduce DGMs and provide a concise mathematical framework for modeling the three most popular approaches: normalizing flows (NF), variational autoencoders (VAE), and generative adversarial networks (GAN). We illustrate the advantages and disadvantages of these basic approaches using numerical experiments. Our goal is to enable and motivate the reader to contribute to this proliferating research area. Our presentation also emphasizes relations between generative modeling and optimal transport.

An introduction to deep generative modeling

A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of variables formula and thus requires invertibility of the mapping and an efficient way to compute the determinant of its Jacobian. To satisfy these requirements, normalizing flows typically consist of carefully chosen components. Continuous normalizing flows (CNFs) are mappings obtained by solving a neural ordinary differential equation (ODE). The neural ODE's dynamics can be chosen almost arbitrarily while ensuring invertibility. Moreover, the log-determinant of the flow's Jacobian can be obtained by integrating the trace of the dynamics' Jacobian along the flow. Our proposed OT-Flow approach tackles two critical computational challenges that limit a more widespread use of CNFs. First, OT-Flow leverages optimal transport (OT) theory to regularize the CNF and enforce straight trajectories that are easier to integrate. Second, OT-Flow features exact trace computation with time complexity equal to trace estimators used in existing CNFs. On five high-dimensional density estimation and generative modeling tasks, OT-Flow performs competitively to state-of-the-art CNFs while on average requiring one-fourth of the number of weights with an 8x speedup in training time and 24x speedup in inference.

OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport.

Differential equations parameterized by neural networks become expensive to solve numerically as training progresses. We propose a remedy that encourages learned dynamics to be easier to solve. Specifically, we introduce a differentiable surrogate for the time cost of standard numerical solvers, using higher-order derivatives of solution trajectories. These derivatives are efficient to compute with Taylor-mode automatic differentiation. Optimizing this additional objective trades model performance against the time cost of solving the learned dynamics. We demonstrate our approach by training substantially faster, while nearly as accurate, models in supervised classification, density estimation, and time-series modelling tasks.

Learning Differential Equations that are Easy to Solve

We compare the discretize-optimize (Disc-Opt) and optimize-discretize (Opt-Disc) approaches for time-series regression and continuous normalizing flows (CNFs) using neural ODEs. Neural ODEs are ordinary differential equations (ODEs) with neural network components. Training a neural ODE is an optimal control problem where the weights are the controls and the hidden features are the states. Every training iteration involves solving an ODE forward and another backward in time, which can require large amounts of computation, time, and memory. Comparing the Opt-Disc and Disc-Opt approaches in image classification tasks, Gholami et al. (2019) suggest that Disc-Opt is preferable due to the guaranteed accuracy of gradients. In this paper, we extend the comparison to neural ODEs for time-series regression and CNFs. Unlike in classification, meaningful models in these tasks must also satisfy additional requirements beyond accurate final-time output, e.g., the invertibility of the CNF. Through our numerical experiments, we demonstrate that with careful numerical treatment, Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs. Disc-Opt reduced costs in six out of seven separate problems with training time reduction ranging from 39% to 97%, and in one case, Disc-Opt reduced training from nine days to less than one day.

Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows.

We propose a neural network approach for solving high-dimensional optimal control problems. In particular, we focus on multi-agent control problems with obstacle and collision avoidance. These problems immediately become high-dimensional, even for moderate phase-space dimensions per agent. Our approach fuses the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizes the value function with a neural network. Our approach yields controls in a feedback form for quick calculation and robustness to moderate disturbances to the system. We train our model using the objective function and optimality conditions of the control problem. Therefore, our training algorithm neither involves a data generation phase nor solutions from another algorithm. Our model uses empirically effective HJB penalizers for efficient training. By training on a distribution of initial states, we ensure the controls' optimality is achieved on a large portion of the state-space. Our approach is grid-free and scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate our approach's effectiveness on a 150-dimensional multi-agent problem with obstacles.

A Neural Network Approach Applied to Multi-Agent Optimal Control

OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport

Epidemic preparedness depends on our ability to predict the trajectory of an epidemic and the human behavior that drives spread in the event of an outbreak. Changes to behavior during an outbreak limit the reliability of syndromic surveillance using large-scale data sources, such as online social media or search behavior, which could otherwise supplement healthcare-based outbreak-prediction methods. Here, we measure behavior change reflected in mobile-phone call-detail records (CDRs), a source of passively collected real-time behavioral information, using an anonymously linked dataset of cell-phone users and their date of influenza-like illness diagnosis during the 2009 H1N1v pandemic. We demonstrate that mobile-phone use during illness differs measurably from routine behavior: Diagnosed individuals exhibit less movement than normal (1.1 to 1.4 fewer unique tower locations; [Formula: see text]), on average, in the 2 to 4 d around diagnosis and place fewer calls (2.3 to 3.3 fewer calls; [Formula: see text]) while spending longer on the phone (41- to 66-s average increase; [Formula: see text]) than usual on the day following diagnosis. The results suggest that anonymously linked CDRs and health data may be sufficiently granular to augment epidemic surveillance efforts and that infectious disease-modeling efforts lacking explicit behavior-change mechanisms need to be revisited.

https://www.pnas.org/content/pnas/118/6/e2005241118.full.pdf

Derek Onken

Papers

OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport.

Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows.

A Neural Network Approach Applied to Multi-Agent Optimal Control

OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport

Cell-phone traces reveal infection-associated behavioral change.