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 machinelearnable structures for scientifically-based learning. We show howUDEs 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 computationallydifficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating climate simulations by 15,000x, can be handled by training UDEs.

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Neural Ordinary Differential Equations (ODEs) are a promising approach to learn dynamical models from time-series data in science and engineering applications. This work aims at learning neural ODEs for stiff systems, which are usually raised from chemical kinetic modeling in chemical and biological systems. We first show the challenges of learning neural ODEs in the classical stiff ODE systems of Robertson’s problem and propose techniques to mitigate the challenges associated with scale separations in stiff systems. We then present successful demonstrations in stiff systems of Robertson’s 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 ODEs. The success of learning stiff neural ODEs opens up possibilities of using neural ODEs in applications with widely varying time-scales, such as chemical dynamics in energy conversion, environmental engineering, and life sciences.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graphbased term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

How do fine modifications to social distancing measures really affect COVID-19 spread? A major problem for health authorities is that we do not know. In an imaginary world, we might develop a harmless biological virus that spreads just like COVID-19, but is traceable via a cheap and reliable diagnosis. By introducing such an imaginary virus into the population and observing how it spreads, we would have a way of learning about COVID-19 because the benign virus would respond to population behaviour and social distancing measures in a similar manner. Such a benign biological virus does not exist. Instead, we propose a safe and privacy-preserving digital alternative. Our solution is to mimic the benign virus by passing virtual tokens between electronic devices when they move into close proximity. As Bluetooth transmission is the most likely method used for such inter-device communication, and as our suggested "virtual viruses" do not harm individuals' software or intrude on privacy, we call these Safe Blues. In contrast to many app-based methods that inform individuals or governments about actual COVID-19 patients or hazards, Safe Blues does not provide information about individuals' locations or contacts. Hence the privacy concerns associated with Safe Blues are much lower than other methods. However, from the point of view of data collection, Safe Blues has two major advantages: - Data about the spread of Safe Blues is uploaded to a central server in real time, which can give authorities a more up-to-date picture in comparison to actual COVID-19 data, which is only available retrospectively. - Sampling of Safe Blues data is not biased by being applied only to people who have shown symptoms or who have come into contact with known positive cases. These features mean that there would be real statistical value in introducing Safe Blues. In the medium term and end game of COVID-19, information from Safe Blues could aid health authorities to make informed decisions with respect to social distancing and other measures. In this paper we outline the general principles of Safe Blues and we illustrate how Safe Blues data together with neural networks may be used to infer characteristics of the progress of the COVID-19 pandemic in real time. Further information is on the Safe Blues website: https://safeblues.org/.

The model-informed drug discovery and development paradigm is now well established among the pharmaceutical industry and regulatory agencies. This success has been mainly due to the ability of pharmacometrics to bring together different modeling strategies, such as population pharmacokinetics/pharmacodynamics (PK/PD) and systems biology/pharmacology. However, there are promising quantitative approaches that are still seldom used by pharmacometricians and that deserve consideration. One such case is the stochastic modeling approach, which can be important when modeling small populations because random events can have a huge impact on these systems. In this review, we aim to raise awareness of stochastic models and how to combine them with existing modeling techniques, with the ultimate goal of making future drug–disease models more versatile and realistic.

Quantitative systems pharmacology (QSP) modeling is applied to address essential questions in drug development, such as the mechanism of action of a therapeutic agent and the progression of disease. Meanwhile, machine learning (ML) approaches also contribute to answering these questions via the analysis of multi-layer ‘omics’ data such as gene expression, proteomics, metabolomics, and high-throughput imaging. Furthermore, ML approaches can also be applied to aspects of QSP modeling. Both approaches are powerful tools and there is considerable interest in integrating QSP modeling and ML. So far, a few successful implementations have been carried out from which we have learned about how each approach can overcome unique limitations of the other. The QSP + ML working group of the International Society of Pharmacometrics QSP Special Interest Group was convened in September, 2019 to identify and begin realizing new opportunities in QSP and ML integration. The working group, which comprises 21 members representing 18 academic and industry organizations, has identified four categories of current research activity which will be described herein together with case studies of applications to drug development decision making. The working group also concluded that the integration of QSP and ML is still in its early stages of moving from evaluating available technical tools to building case studies. This paper reports on this fast-moving field and serves as a foundation for future codification of best practices.