Chris Rackauckas

Francesco Martinuzzi, Chris Rackauckas, Anas Abdelrehim et al.

We introduce ReservoirComputing.jl, an open source Julia library for reservoir computing models. The software offers a great number of algorithms presented in the literature, and allows to expand on them with both internal and external tools in a simple way. The implementation is highly modular, fast and comes with a comprehensive documentation, which includes reproduced experiments from literature. The code and documentation are hosted on Github under an MIT license https://github.com/SciML/ReservoirComputing.jl.

Gaurav Arya, Moritz Schauer, Frank Schäfer et al.

Automatic differentiation (AD), a technique for constructing new programs which compute the derivative of an original program, has become ubiquitous throughout scientific computing and deep learning due to the improved performance afforded by gradient-based optimization. However, AD systems have been restricted to the subset of programs that have a continuous dependence on parameters. Programs that have discrete stochastic behaviors governed by distribution parameters, such as flipping a coin with probability $p$ of being heads, pose a challenge to these systems because the connection between the result (heads vs tails) and the parameters ($p$) is fundamentally discrete. In this paper we develop a new reparameterization-based methodology that allows for generating programs whose expectation is the derivative of the expectation of the original program. We showcase how this method gives an unbiased and low-variance estimator which is as automated as traditional AD mechanisms. We demonstrate unbiased forward-mode AD of discrete-time Markov chains, agent-based models such as Conway's Game of Life, and unbiased reverse-mode AD of a particle filter. Our code package is available at https://github.com/gaurav-arya/StochasticAD.jl.

Chaopeng Shen, Alison P. Appling, Pierre Gentine et al.

Process-Based Modeling (PBM) and Machine Learning (ML) are often perceived as distinct paradigms in the geosciences. Here we present differentiable geoscientific modeling as a powerful pathway toward dissolving the perceived barrier between them and ushering in a paradigm shift. For decades, PBM offered benefits in interpretability and physical consistency but struggled to efficiently leverage large datasets. ML methods, especially deep networks, presented strong predictive skills yet lacked the ability to answer specific scientific questions. While various methods have been proposed for ML-physics integration, an important underlying theme -- differentiable modeling -- is not sufficiently recognized. Here we outline the concepts, applicability, and significance of differentiable geoscientific modeling (DG). "Differentiable" refers to accurately and efficiently calculating gradients with respect to model variables, critically enabling the learning of high-dimensional unknown relationships. DG refers to a range of methods connecting varying amounts of prior knowledge to neural networks and training them together, capturing a different scope than physics-guided machine learning and emphasizing first principles. Preliminary evidence suggests DG offers better interpretability and causality than ML, improved generalizability and extrapolation capability, and strong potential for knowledge discovery, while approaching the performance of purely data-driven ML. DG models require less training data while scaling favorably in performance and efficiency with increasing amounts of data. With DG, geoscientists may be better able to frame and investigate questions, test hypotheses, and discover unrecognized linkages.

Avik Pal, Alan Edelman, Chris Rackauckas

Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a search over a space of plausible dynamical systems. However, controlling the computational cost for these models is difficult since it relies on the number of steps the adaptive solver takes. Most prior works have used higher-order methods to reduce prediction timings while greatly increasing training time or reducing both training and prediction timings by relying on specific training algorithms, which are harder to use as a drop-in replacement due to strict requirements on automatic differentiation. In this manuscript, we use internal cost heuristics of adaptive differential equation solvers at stochastic time points to guide the training toward learning a dynamical system that is easier to integrate. We "close the black-box" and allow the use of our method with any adjoint technique for gradient calculations of the differential equation solution. We perform experimental studies to compare our method to global regularization to show that we attain similar performance numbers without compromising the flexibility of implementation on ordinary differential equations (ODEs) and stochastic differential equations (SDEs). We develop two sampling strategies to trade off between performance and training time. Our method reduces the number of function evaluations to 0.556-0.733x and accelerates predictions by 1.3-2x.

Vinicius V. Santana, Erbet Costa, Carine M. Rebello et al.

This study presents a systematic machine learning approach for creating efficient hybrid models and discovering sorption uptake models in non-linear advection-diffusion-sorption systems. It demonstrates an effective method to train these complex systems using gradient based optimizers, adjoint sensitivity analysis, and JIT-compiled vector Jacobian products, combined with spatial discretization and adaptive integrators. Sparse and symbolic regression were employed to identify missing functions in the artificial neural network. The robustness of the proposed method was tested on an in-silico data set of noisy breakthrough curve observations of fixed-bed adsorption, resulting in a well-fitted hybrid model. The study successfully reconstructed sorption uptake kinetics using sparse and symbolic regression, and accurately predicted breakthrough curves using identified polynomials, highlighting the potential of the proposed framework for discovering sorption kinetic law structures.

Mohamed Tarek, Jose Storopoli, Casey Davis et al.

This paper provides a comprehensive tutorial for Bayesian practitioners in pharmacometrics using Pumas workflows. We start by giving a brief motivation of Bayesian inference for pharmacometrics highlighting limitations in existing software that Pumas addresses. We then follow by a description of all the steps of a standard Bayesian workflow for pharmacometrics using code snippets and examples. This includes: model definition, prior selection, sampling from the posterior, prior and posterior simulations and predictions, counter-factual simulations and predictions, convergence diagnostics, visual predictive checks, and finally model comparison with cross-validation. Finally, the background and intuition behind many advanced concepts in Bayesian statistics are explained in simple language. This includes many important ideas and precautions that users need to keep in mind when performing Bayesian analysis. Many of the algorithms, codes, and ideas presented in this paper are highly applicable to clinical research and statistical learning at large but we chose to focus our discussions on pharmacometrics in this paper to have a narrower scope in mind and given the nature of Pumas as a software primarily for pharmacometricians.

Sebastian Micluta-Campeanu, Avinash Subramanian, Anas Abdelrehim et al.

Calibration of dynamic models to data is an important step in building building digital twins of HVAC equipment, thermal loads and control systems. Sometimes, when a model fails to calibrate to data, a possible cause is that the model has made too many sim- plifying assumptions and is missing physics. In this paper we propose a semi-automated approach, called Dyad Model Discovery, that can augment the physical equations of the model with symbolic expressions discovered from the data. We demonstrate this method on a digital twin of a transportation refrigeration unit to improve its predictive performance, trained using telemetry data. An engineer-in-the-loop workflow is proposed, which provides suggestions to the user which can then be accepted or rejected. This is the first AI-assisted engineering design workflow to our knowledge.

Raphaël Pestourie, Youssef Mroueh, Chris Rackauckas et al.

Many physics and engineering applications demand Partial Differential Equations (PDE) property evaluations that are traditionally computed with resource-intensive high-fidelity numerical solvers. Data-driven surrogate models provide an efficient alternative but come with a significant cost of training. Emerging applications would benefit from surrogates with an improved accuracy-cost tradeoff, while studied at scale. Here we present a "physics-enhanced deep-surrogate" ("PEDS") approach towards developing fast surrogate models for complex physical systems, which is described by PDEs. Specifically, a combination of a low-fidelity, explainable physics simulator and a neural network generator is proposed, which is trained end-to-end to globally match the output of an expensive high-fidelity numerical solver. Experiments on three exemplar testcases, diffusion, reaction-diffusion, and electromagnetic scattering models, show that a PEDS surrogate can be up to 3$\times$ more accurate than an ensemble of feedforward neural networks with limited data ($\approx 10^3$ training points), and reduces the training data need by at least a factor of 100 to achieve a target error of 5%. Experiments reveal that PEDS provides a general, data-driven strategy to bridge the gap between a vast array of simplified physical models with corresponding brute-force numerical solvers modeling complex systems, offering accuracy, speed, data efficiency, as well as physical insights into the process.

Frank Schäfer, Mohamed Tarek, Lyndon White et al.

No single Automatic Differentiation (AD) system is the optimal choice for all problems. This means informed selection of an AD system and combinations can be a problem-specific variable that can greatly impact performance. In the Julia programming language, the major AD systems target the same input and thus in theory can compose. Hitherto, switching between AD packages in the Julia Language required end-users to familiarize themselves with the user-facing API of the respective packages. Furthermore, implementing a new, usable AD package required AD package developers to write boilerplate code to define convenience API functions for end-users. As a response to these issues, we present AbstractDifferentiation.jl for the automatized generation of an extensive, unified, user-facing API for any AD package. By splitting the complexity between AD users and AD developers, AD package developers only need to implement one or two primitive definitions to support various utilities for AD users like Jacobians, Hessians and lazy product operators from native primitives such as pullbacks or pushforwards, thus removing tedious -- but so far inevitable -- boilerplate code, and enabling the easy switching and composing between AD implementations for end-users.

Yingbo Ma, Shashi Gowda, Ranjan Anantharaman et al.

Getting good performance out of numerical equation solvers requires that the user has provided stable and efficient functions representing their model. However, users should not be trusted to write good code. In this manuscript we describe ModelingToolkit (MTK), a symbolic equation-based modeling system which allows for composable transformations to generate stable, efficient, and parallelized model implementations. MTK blurs the lines of traditional symbolic computing by acting directly on a user's numerical code. We show the ability to apply graph algorithms for automatically parallelizing and performing index reduction on code written for differential-algebraic equation (DAE) solvers, "fixing" the performance and stability of the model without requiring any changes to on the user's part. We demonstrate how composable model transformations can be combined with automated data-driven surrogate generation techniques, allowing machine learning methods to generate accelerated approximate models within an acausal modeling framework. These reduced models are shown to outperform the Dymola Modelica compiler on an HVAC model by 590x at 3\% error. Together, this demonstrates MTK as a system for bringing the latest research in graph transformations directly to modeling applications.

Raj Dandekar, Karen Chung, Vaibhav Dixit et al.

Recently, Neural Ordinary Differential Equations has emerged as a powerful framework for modeling physical simulations without explicitly defining the ODEs governing the system, but instead learning them via machine learning. However, the question: "Can Bayesian learning frameworks be integrated with Neural ODE's to robustly quantify the uncertainty in the weights of a Neural ODE?" remains unanswered. In an effort to address this question, we primarily evaluate the following categories of inference methods: (a) The No-U-Turn MCMC sampler (NUTS), (b) Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) and (c) Stochastic Langevin Gradient Descent (SGLD). We demonstrate the successful integration of Neural ODEs with the above Bayesian inference frameworks on classical physical systems, as well as on standard machine learning datasets like MNIST, using GPU acceleration. On the MNIST dataset, we achieve a posterior sample accuracy of 98.5% on the test ensemble of 10,000 images. Subsequently, for the first time, we demonstrate the successful integration of variational inference with normalizing flows and Neural ODEs, leading to a powerful Bayesian Neural ODE object. Finally, considering a predator-prey model and an epidemiological system, we demonstrate the probabilistic identification of model specification in partially-described dynamical systems using universal ordinary differential equations. Together, this gives a scientific machine learning tool for probabilistic estimation of epistemic uncertainties.

Ranjan Anantharaman, Yingbo Ma, Shashi Gowda et al.

Modern design, control, and optimization often requires simulation of highly nonlinear models, leading to prohibitive computational costs. These costs can be amortized by evaluating a cheap surrogate of the full model. Here we present a general data-driven method, the continuous-time echo state network (CTESN), for generating surrogates of nonlinear ordinary differential equations with dynamics at widely separated timescales. We empirically demonstrate near-constant time performance using our CTESNs on a physically motivated scalable model of a heating system whose full execution time increases exponentially, while maintaining relative error of within 0.2 %. We also show that our model captures fast transients as well as slow dynamics effectively, while other techniques such as physics informed neural networks have difficulties trying to train and predict the highly nonlinear behavior of these models.

Mike Innes, Alan Edelman, Keno Fischer et al.

Scientific computing is increasingly incorporating the advancements in machine learning and the ability to work with large amounts of data. At the same time, machine learning models are becoming increasingly sophisticated and exhibit many features often seen in scientific computing, stressing the capabilities of machine learning frameworks. Just as the disciplines of scientific computing and machine learning have shared common underlying infrastructure in the form of numerical linear algebra, we now have the opportunity to further share new computational infrastructure, and thus ideas, in the form of Differentiable Programming. We describe Zygote, a Differentiable Programming system that is able to take gradients of general program structures. We implement this system in the Julia programming language. Our system supports almost all language constructs (control flow, recursion, mutation, etc.) and compiles high-performance code without requiring any user intervention or refactoring to stage computations. This enables an expressive programming model for deep learning, but more importantly, it enables us to incorporate a large ecosystem of libraries in our models in a straightforward way. We discuss our approach to automatic differentiation, including its support for advanced techniques such as mixed-mode, complex and checkpointed differentiation, and present several examples of differentiating programs.

Chris Rackauckas, Mike Innes, Yingbo Ma et al.

DiffEqFlux.jl is a library for fusing neural networks and differential equations. In this work we describe differential equations from the viewpoint of data science and discuss the complementary nature between machine learning models and differential equations. We demonstrate the ability to incorporate DifferentialEquations.jl-defined differential equation problems into a Flux-defined neural network, and vice versa. The advantages of being able to use the entire DifferentialEquations.jl suite for this purpose is demonstrated by counter examples where simple integration strategies fail, but the sophisticated integration strategies provided by the DifferentialEquations.jl library succeed. This is followed by a demonstration of delay differential equations and stochastic differential equations inside of neural networks. We show high-level functionality for defining neural ordinary differential equations (neural networks embedded into the differential equation) and describe the extra models in the Flux model zoo which includes neural stochastic differential equations. We conclude by discussing the various adjoint methods used for backpropogation of the differential equation solvers. DiffEqFlux.jl is an important contribution to the area, as it allows the full weight of the differential equation solvers developed from decades of research in the scientific computing field to be readily applied to the challenges posed by machine learning and data science.