ModelingToolkit: A Composable Graph Transformation System For Equation-Based Modeling (2103.05244v3)

Abstract: 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.

• The paper presents a composable Julia framework that automates graph-based transformations to convert higher-index DAEs into index-1 forms using the Pantelides algorithm.
• The paper demonstrates enhanced solver performance by extracting parallelism through strongly connected component analysis and integrating data-driven surrogates.
• The paper highlights practical benefits by reducing manual model optimization and accelerating computations up to 590x in applications like HVAC and pendulum systems.

ModelingToolkit: A Composable Graph Transformation System for Equation-Based Modeling

The paper presents the development and application of ModelingToolkit (MTK), a Julia-based modeling framework designed to address inefficiencies in equation-based modeling systems. MTK facilitates the automation of transformations necessary for improving the performance and stability of models governed by differential-algebraic equations (DAEs). The authors propose a novel approach for bridging symbolic computing and numerical paradigms, leveraging graph-based transformations to optimize code execution and obviate the need for users to manually refine their code for solver readiness.

Central to the contribution of MTK is its ability to operate directly within the numerical code environment provided by the Julia language, thereby sidestepping the limitations inherent in traditional declarative modeling environments like Modelica. This seamless integration allows users to maintain fidelity to imperative programming practices while still gaining the stability and performance benefits afforded by symbolic transformations.

One of the key features of MTK is its capability to automatically perform index reduction on DAEs using Pantelides algorithm and other extension methodologies. This automated process alleviates the user from manually adjusting the formulation of DAEs, which is often error-prone and time-intensive. By transforming higher index DAEs into index-1 forms, MTK enhances solver performance and improves the numerical solvability of the given system, as illustrated with the pendulum case paper in the paper.

MTK also integrates parallel computing into its framework, extracting parallelism from the structure of DAEs during the tearing process. By identifying strongly connected components within the system graph, MTK determines which algebraic equations can be solved concurrently, optimizing the computational resources utilized and reducing computation time significantly, as seen in the scalable examples like the HVAC model.

Moreover, MTK supports data-driven model order reduction techniques, providing a robust framework for embedding machine learning-generated surrogates within the modeling ecosystem. This functionality is exemplified by the surrogatize transformation which utilizes continuous-time echo state networks (CTESN) to produce reduced-order models. Such models maintain high fidelity with original complex systems while dramatically decreasing computation time, providing a performance acceleration by a factor of up to 590x compared to high-fidelity solvers, as demonstrated on the HVAC model.

The integration of these features into a single composable system offers significant implications for the field of computational modeling. MTK's incorporation of symbolic transformations and graph-based algorithms directly into user code represents a substantial step towards reducing the burden of manual model optimization. By enabling user-driven yet automated compositional transformation chains, MTK creates a flexible and powerful environment for engaging with both established numerical techniques and emerging data-driven approaches.

Looking forward, MTK sets the stage for further innovation in automated modeling systems. The capability to embed new symbolic transformations and analysis routines will empower researchers to extend the functionality and applicability of the toolkit, potentially exploring such areas as floating-point accuracy optimization, delay differential equations handling, and structural analysis for parameter identifiability. The system's open-source nature invites collaboration from the broader scientific community, fostering advancements at the intersection of symbolic computation and numerical simulation.