Diff Grok

A lightweight TypeScript library for solving initial value problem (IVP) for ordinary differential equations (ODEs) using numerical methods. This library focuses on solving stiff equations.

Features

• Solving both stiff and non-stiff equations
• Fast computations
• A set of Rosenbrock–Wanner-type methods:
  • The modified Rosenbrock triple (MRT)
  • The ROS3PRw method
  • The ROS34PRw method
• Scripting:
  • declarative specification of models
  • auto-generated JavaScript code
• Integration with the Datagrok platform
• Zero dependencies

Installation

npm i diff-grok
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Solving

General

To find numerical solution of a problem:

$$\frac{dy}{dt} = f(t, y)$$ $$y(t_{0}) = y_0$$

on the segment $[t_0, t_1]$ with the step $h$:

1. Import ODEs and a desired numerical method:

   • mrt - the MRT method
   • ros3prw- the ROS3PRw method
   • ros34prw - the ROS34PRw method

2. Specify ODEs object that defines a problem:

   • name - name of a model

   • arg - independent variable specification. This is in object with fields:

     • name - name of the argument, $t$
     • start - initial value of the argument, $t_0$
     • finish - final value of the argument, $t_1$
     • step - solution grid step, $h$

   • initial - initial values, $y_0$

   • func - right-hand side of the system, $f(t, y)$. This is a function (t: number, y: Float64Array, output: Float64Array) => void:

     • t - value of independent variable $t$
     • y - values of $y$
     • output - output values of $f(t, y)$

   • tolerance - numerical tolerance

   • solutionColNames - names of solutions, i.e. names of the vector $y$ elements

3. Call numerical method. It returns Float64Array-arrays with values of an argument and approximate solutions.

Diff Grok is designed to provide fast computations. Check performance for the details.

Example

Consider the following problem:

$$\begin{cases} \frac{dx}{dt} = x + y - t \ \frac{dy}{dt} = x y + t \ x(0) = 1 \ y(0) = -1 \end{cases}$$

To solve it on the segment $[0, 2]$ with the step $0.01$ using the MRT method with the tolerance $10^{-7}$, we start with imports:

import {ODEs, mrt} from 'diff-grok';
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Next, we create

const task: ODEs = {
    name: 'Example', // name of your model
    arg: {
        name: 't',  // name of the argument
        start: 0,   // initial value of the argument
        finish: 2,  // final value of the argument
        step: 0.01, // solution grid step
    },
    initial: [1, -1], // initial values
    func: (t: number, y: Float64Array, output: Float64Array) => { // right-hand side of the system
      output[0] = y[0] + y[1] - t; // 1-st equation
      output[1] = y[0] * y[1] + t; // 2-nd equation
    },
    tolerance: 1e-7, // tolerance
    solutionColNames: ['x', 'y'], // names of solution functions
};
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Finally, we call the specified numerical method to solve task:

const solution = mrt(task);
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Currently, solution contains:

• solution[0] - values of $t$, i.e. the range $0..2$ with the step $0.01$
• solution[1] - values of $x(t)$ at the points of this range
• solution[2] - values of $y(t)$ at the points of the same range

Find this example in basic-use.ts.

Performance

The following classic problems are used to evaluate efficiency of Diff Grok methods:

• Rober
  • a stiff system of 3 nonlinear ODEs
  • describes the kinetics of an autocatalytic reaction given by Robertson
  • robertson.ts
• HIRES
  • a stiff system of 8 non-linear equations
  • explains the `High Irradiance Responses' (HIRES) of photomorphogenesis on the basis of phytochrome, by means of a chemical reaction involving eight reactants
  • hires.ts
• VDPOL
  • a system of 2 ODEs proposed by B. van der Pol
  • describes the behaviour of nonlinear vacuum tube circuits
  • vdpol.ts
• OREGO
  • a stiff system of 3 non-linear equations
  • simulates Belousov-Zhabotinskii reaction
  • orego.ts
• E5
  • a stiff system of 4 non-linear ODEs
  • represents a chemical pyrolysis model
  • e5.ts
• Pollution
  • a stiff system of 20 non-linear equations
  • describes a chemical reaction part of the air pollution model designed at The Dutch National Institute of Public Health and Environmental Protection
  • pollution.ts

The MRT, ROS3PRw and ROS34PRw methods demonstrate the following time performance (AMD Ryzen 5 5600H 3.30 GHz CPU):

Problem	Segment	Points	Tolerance	MRT, ms	ROS3PRw, ms	ROS34PRw, ms
Rober	[0, 10E+11]	40K	1E-7	103	446	285
HIRES	[0, 321.8122]	32K	1E-10	222	362	215
VDPOL	[0, 2000]	20K	1E-12	963	1576	760
OREGO	[0, 360]	36K	1E-8	381	483	199
E5	[0, 10E+13]	40K	1E-6	14	17	8
Pollution	[0, 60]	30K	1E-6	36	50	23

Run check-methods.ts to check results.

Scripting

The library provides tools for declarative specifying models defined by IVPs. This feature enables a development of "no-code" modeling tools seamlessly integrated with the Datagrok platform.

Model components and syntax

Each model has a simple declarative syntax.

Core blocks

These blocks define the basic mathematical model and are required for any model:

1. #name: Add a model identifier

   #name: Problem
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2. #equations: Define the system of ODEs to solve. Diff Grok supports any number of equations with single or multi-letter variable names

   #equations:
     dx/dt = x + y + exp(t)
     dy/dt = x - y - cos(t)
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3. #argument: Defines

   • independent variable
   • its initial value (initial)
   • final value (final), and
   • grid step (step)

   The solver calculates values at each step interval across the specified [initial,final] range.

   #argument: t
     initial = 0
     final = 1
     step = 0.01
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4. #inits: Defines initial values for functions being solved

   #inits:
     x = 2
     y = 5
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Comments

• #comment: Write a comment in any place of your model

  #comment:
    You can provide any text here. The lib ignores it.
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• Place comments right in formulas using //

  #equations:
    dx/dt = x + y + exp(t) // 1-st equation
    dy/dt = x - y - cos(t) // 2-nd equation
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Model parameters

These blocks define values used in equations. Choose type based on intended use:

• #parameters: Generate UI controls for model exploration

  #parameters:
    P1 = 1
    P2 = -1
  
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• #constants: Use for fixed values in equations that don't require UI controls

  #constants:
    C1 = 1
    C2 = 3
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Auxiliary calculations

This block defines mathematical functions using #parameters, #constants, #argument, and other functions. These are direct calculations (no ODEs involved). Use them to break down complex calculations and simplify your equations.

• #expressions

  #expressions:
    E1 = C1 * t + P1
    E2 = C2 * cos(2 * t) + P2
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JS-code generation

To transform any model to JavaScript code with an appropriate specification of ODEs object, follow the steps:

1. Import the parsing and code generating tools:

import {getIVP, getJScode} from 'diff-grok';
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2. Define a string with a model specification, use a simple model syntax:

const model = `
#name: Example
#equations:
  dx/dt = x + y - cos(t)
  dy/dt = x - y + sin(t)
...
`;
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3. Parse formulas:

const ivp = getIVP(model);
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The method getIVP parses formulas and returns IVP object specifying a model.

4. Generate JS-code:

const lines = getJScode(ivp);
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The method getJScode transforms IVP object to JavaScript code. It returns an array of strings with this code.

Find this example in scripting.ts.

Pipelines

Diff Grok pipeline is a powerful feature for complex process simulation and model analysis in webworkers. It wraps the main solver with a set of actions that perform pre- and post-processing of a model inputs & outputs. In addition, they provide an output customization.

1. Start with imports:

import * as DGL from 'diff-grok';
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2. Define your model:

const model = `#name: My model
#equations:
  dx/dt = ...
  dy/dt = ...
  ...

#inits:
  x = 2
  y = 3
  ...
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3. Generate IVP-objects:

• for the main thread computations:

const ivp = DGL.getIVP(model);
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• for computations in workers:

const ivpWW = DGL.getIvp2WebWorker(ivp);
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4. Set model inputs:

const inputs = {
  x: 2,
  y: 30,
  ...
};
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5. Create typed input array:

const inputVector = DGL.getInputVector(inputs, ivp);
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6. Get a pipeline:

const creator = DGL.getPipelineCreator(ivp);
const pipeline = creator.getPipeline(inputVector);
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You can pass pipeline, ivpWW, and inputVector to webworkers.

7. Apply pipeline to perform computations:

const solution = DGL.applyPipeline(pipeline, ivpWW, inputVector);
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Find complete examples in these files:

• pipeline-use.ts - A basic example demonstrating pipeline usage
• model-updates.ts - A simulation of a multi-stage process using pipelines
• cyclic-model.ts - A simulation of a cyclic process using pipelines

Integration with Datagrok

Datagrok is a platform enabling powerful scientific computing capabilities. It provides next-generation environment for leveraging interactive visualizations, data access, machine learning, and enterprise features to enable developing, publishing, discovering, and using scientific applications.

The library is seamlessly integrated to Datagrok via the Diff Studio package. It provides

• Numerical solving IVPs directly in the browser
• "No-code" models development
• Solving both stiff and non-stiff systems of ODEs
• Automatic generation of user interfaces
• Interactive visualization and model exploration
• Sensitivity analysis and parameters optimization
• Sharing models and computational results

Run the Diff Studio app and check interactive modeling:

Learn more

• Diff Studio application docs
• Diff Studio example models
• Parameters optimization
• Sensitivity analysis