Simulation Solvers & Numerical Integrators

ModelScript embeds a multi-tier suite of numerical solvers in @modelscript/simulate to support both desktop simulations and massive batched cloud/WebGPU parameter sweeps.

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Supported Solvers

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1. SUNDIALS CVODE & IDA Integrators

For high-precision industrial models, ModelScript integrates Lawrence Livermore National Laboratory's SUNDIALS suite compiled to WebAssembly:

• CVODE: Solves Initial Value Problems (IVPs) for stiff and non-stiff systems of ordinary differential equations (ODEs). Uses Variable-Order Variable-Step-size Adams-Moulton methods for non-stiff systems and Backward Differentiation Formulas (BDF) for stiff systems.
• IDA: Solves Initial Value Problems for systems of differential-algebraic equations (DAEs) of index up to 1:$$F(t,y,y^{\prime })=0$$Includes dense and sparse linear solvers with adaptive step-size control.

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2. WebGPU Batched Parallel Solvers

When performing Monte Carlo uncertainty analysis or hyperparameter sweeps, sequential simulation is a bottleneck.

ModelScript compiles linearized DAE blocks into WebGPU compute shaders (@modelscript/simulate/gpu):

• Runs thousands of parameter combinations in parallel on GPU hardware.
• Scales efficiently across client GPUs in browsers and headless datacenter instances.
• Returns statistical distributions, mean trajectories, and variance envelopes directly.

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3. Direct Collocation Optimal Control

ModelScript formulates dynamic optimization and optimal control problems directly from Modelica models:

$$\underset{u(t)}{min}{\int }_0^{T}L(x(t),u(t),t)dt$$$$\text{subject to }F(t,x,\dot{x},u)=0,u_{min}\le u(t)\le u_{max}$$

• Transcription: Transcribes the continuous system into non-linear programming (NLP) problems using Radau or Lobatto collocation points.
• Solvers: Solves the resulting sparse NLP using interior-point optimization techniques.

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4. Surrogate Reduced Order Models (ROM)

High-fidelity 3D simulations (such as CFD or FEA) are often too computationally expensive for real-time control loops or web deployment.

ModelScript's surrogate modeling pipeline (msx surrogate):

1. Sampling: Generates experimental design samples (Latin Hypercube, Sobol sequences).
2. Execution: Evaluates high-fidelity runs across parameter spaces.
3. Training: Trains compact surrogate models using:
   • Polynomial Chaos Expansion (PCE)
   • Gaussian Process Regression (Kriging)
   • Neural Ordinary Differential Equations (Neural ODEs)
4. Embedding: Exports the trained surrogate as a standardized FMI Functional Mock-up Unit (FMU) or standalone WebAssembly module.