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Open source Differentiable ODE Solving Infrastructure

Rakshit Kr. Singh, Aaron Rock Menezes, Rida Irfan, Bharath Ramsundar

TL;DR

This work integrates GPU-accelerated ODE solvers into the open-source DeepChem framework, making these tools easily accessible and demonstrating high accuracy with mean squared errors and scalability in solving large systems with up to 100 compartments.

Abstract

Ordinary Differential Equations (ODEs) are widely used in physics, chemistry, and biology to model dynamic systems, including reaction kinetics, population dynamics, and biological processes. In this work, we integrate GPU-accelerated ODE solvers into the open-source DeepChem framework, making these tools easily accessible. These solvers support multiple numerical methods and are fully differentiable, enabling easy integration into more complex differentiable programs. We demonstrate the capabilities of our implementation through experiments on Lotka-Volterra predator-prey dynamics, pharmacokinetic compartment models, neural ODEs, and solving PDEs using reaction-diffusion equations. Our solvers achieved high accuracy with mean squared errors ranging from $10^{-4}$ to $10^{-6}$ and showed scalability in solving large systems with up to 100 compartments.

Open source Differentiable ODE Solving Infrastructure

TL;DR

This work integrates GPU-accelerated ODE solvers into the open-source DeepChem framework, making these tools easily accessible and demonstrating high accuracy with mean squared errors and scalability in solving large systems with up to 100 compartments.

Abstract

Ordinary Differential Equations (ODEs) are widely used in physics, chemistry, and biology to model dynamic systems, including reaction kinetics, population dynamics, and biological processes. In this work, we integrate GPU-accelerated ODE solvers into the open-source DeepChem framework, making these tools easily accessible. These solvers support multiple numerical methods and are fully differentiable, enabling easy integration into more complex differentiable programs. We demonstrate the capabilities of our implementation through experiments on Lotka-Volterra predator-prey dynamics, pharmacokinetic compartment models, neural ODEs, and solving PDEs using reaction-diffusion equations. Our solvers achieved high accuracy with mean squared errors ranging from to and showed scalability in solving large systems with up to 100 compartments.

Paper Structure

This paper contains 37 sections, 8 equations, 8 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Prior Research at solving ODE Systems
  4. Numerical Methods for solving ODE systems
  5. Euler's Method:
  6. Runge-Kutta Methods:
  7. Predictor-Corrector Methods:
  8. Multi-step Methods:
  9. Implicit Methods:
  10. Understanding Differentiable Programming
  11. Ordinary Differential Equations (ODEs) in Systems Biology
  12. Related Work
  13. Methodology
  14. DeepChem and Differentiable Optimizers
  15. Using DeepChem for System Identification
  16. ...and 22 more sections

Figures (8)

  • Figure 1: DeepChem ODE Solving workflow
  • Figure 2: DeepChem parameter estimation workflow
  • Figure 3: Predator-prey equation solutions obtained using the SciPy and DeepChem match closely.
  • Figure 4: Schematic representation of a three-compartment model without a direct connection between peripheral compartments.
  • Figure 5: Harmonic Oscillator Dynamics using Neural ODEs: We compare a simulated Damped Harmonic Oscillator, solved using an ODE Solver, with the predictions of a Neural ODE and use the model to predict the dynamics of the system for the next 30 seconds.
  • ...and 3 more figures