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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.

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Implicit deep learning architectures, like Neural ODEs and Deep Equilibrium Models (DEQs), separate the definition of a layer from the description of its solution process. While implicit layers allow features such as depth to adapt to new scenarios and inputs automatically, this adaptivity makes its computational expense challenging to predict. Numerous authors have noted that implicit layer techniques can be more computationally intensive than explicit layer methods. In this manuscript, we address the question: is there a way to simultaneously achieve the robustness of implicit layers while allowing the reduced computational expense of an explicit layer? To solve this we develop Skip DEQ, an implicit-explicit (IMEX) layer that simultaneously trains an explicit prediction followed by an implicit correction. We show that training this explicit layer is free and even decreases the training time by 2.5x and prediction time by 3.4x. We then further increase the "implicitness" of the DEQ by redefining the method in terms of an infinite time neural ODE which paradoxically decreases the training cost over a standard neural ODE by not requiring backpropagation through time. We demonstrate how the resulting Continuous Skip DEQ architecture trains more robustly than the original DEQ while achieving faster training and prediction times. Together, this manuscript shows how bridging the dichotomy of implicit and explicit deep learning can combine the advantages of both techniques.

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Shashi Gowda, Yingbo Ma, Alessandro Cheli, Maja Gwozdz, Viral B. Shah, Alan Edelman, Christopher Rackauckas

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. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase 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.

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Rachel C. Kurchin, Eric Muckley, Lance Kavalsky, Vinay Hegde, Dhairya Gandhi, Xiaoyu Sun, Matthew Johnson, Alan Edelman, James Saal, Christopher Vincent Rackauckas, Bryce Meredig, Viral Shah, Venkatasubramanian Viswanathan

Large-scale electrification is vital to addressing the climate crisis, but many engineering challenges remain to fully electrifying both the chemical industry and transportation. In both of these areas, new electrochemical materials and systems will be critical, but developing these systems currently relies heavily on computationally expensive first-principles simulations as well as human-time-intensive experimental trial and error. We propose to develop an automated workflow that accelerates these computational steps by introducing both automated error handling in generating the first-principles training data as well as physics-informed machine learning surrogates to further reduce computational cost. It will also have the capacity to include automated experiments "in the loop" in order to dramatically accelerate the overall materials discovery pipeline.

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Ranjan Anantharaman, Yingbo Ma, Shashi Gowda, Chris Laughman, Viral Shah, Alan Edelman, Chris Rackauckas

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.

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Albert R. Gnadt, Joseph Belarge, Aaron Canciani, Lauren Conger, Joseph Curro, Alan Edelman, Peter Morales, Michael F. O'Keeffe, Jonathan Taylor, Christopher Rackauckas

Harnessing the magnetic field of the earth for navigation has shown promise as a viable alternative to other navigation systems. A magnetic navigation system collects its own magnetic field data using a magnetometer and uses magnetic anomaly maps to determine the current location. The greatest challenge with magnetic navigation arises when the magnetic field data from the magnetometer on the navigation system encompass the magnetic field from not just the earth, but also from the vehicle on which it is mounted. It is difficult to separate the earth magnetic anomaly field magnitude, which is crucial for navigation, from the total magnetic field magnitude reading from the sensor. The purpose of this challenge problem is to decouple the earth and aircraft magnetic signals in order to derive a clean signal from which to perform magnetic navigation. Baseline testing on the dataset shows that the earth magnetic field can be extracted from the total magnetic field using machine learning (ML). The challenge is to remove the aircraft magnetic field from the total magnetic field using a trained neural network. These challenges offer an opportunity to construct an effective neural network for removing the aircraft magnetic field from the dataset, using an ML algorithm integrated with physics of magnetic navigation.

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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.

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Convolutions have long been regarded as fundamental to applied mathematics, physics and engineering. Their mathematical elegance allows for common tasks such as numerical differentiation to be computed efficiently on large data sets. Efficient computation of convolutions is critical to artificial intelligence in real-time applications, like machine vision, where convolutions must be continuously and efficiently computed on tens to hundreds of kilobytes per second. In this paper, we explore how convolutions are used in fundamental machine vision applications. We present an accelerated n-dimensional convolution package in the high performance computing language, Julia, and demonstrate its efficacy in solving the time to contact problem for machine vision. Results are measured against synthetically generated videos and quantitatively assessed according to their mean squared error from the ground truth. We achieve over an order of magnitude decrease in compute time and allocated memory for comparable machine vision applications. All code is packaged and integrated into the official Julia Package Manager to be used in various other scenarios.

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