Shashi Gowda

Shashi Gowda, Yingbo Ma, Alessandro Cheli et al.

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.

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.

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.