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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-graph-based 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 demonstratemore »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.« less

Modern design, control, and optimization often require multiple expensive simulations of highly nonlinear stiff models. These costs can be amortized by training a cheap surrogate of the full model, which can then be used repeatedly. 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 the ability to accelerate a physically motivated scalable model of a heating system by 98x while maintaining relative error of within 0.2 %. We showcase the ability for this surrogate to accurately handle highly stiff systems which have been shown to cause training failures with common surrogate methods such as Physics-Informed Neural Networks (PINNs), Long Short Term Memory (LSTM) networks, and discrete echo state networks (ESN). We show that our model captures fast transients as well as slow dynamics, while demonstrating that fixed time step machine learning techniques are unable to adequately capture the multi-rate behavior. Together this provides compelling evidence for the ability of CTESN surrogates to predict and accelerate highly stiff dynamical systems which are unable to be directly handled by previous scientific machine learning techniques.