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Related Concept Videos

State Space Representation01:27

State Space Representation

496
The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
496
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

314
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
314
Transfer Function to State Space01:23

Transfer Function to State Space

727
State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
727
State Space to Transfer Function01:21

State Space to Transfer Function

533
The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
533
Multi-input and Multi-variable systems01:22

Multi-input and Multi-variable systems

371
Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...
371
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

255
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
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State-space models are accurate and efficient neural operators for dynamical systems.

Zheyuan Hu1, Nazanin Ahmadi Daryakenari2, Qianli Shen1

  • 1Department of Computer Science, National University of Singapore, 119077, Singapore.

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Summary

Mamba, a new state-space model, excels at learning dynamical systems. It offers superior accuracy and efficiency, especially in challenging extrapolation tasks, outperforming existing methods in scientific machine learning.

Keywords:
Dynamical systemMambaNeural operatorState-space model

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Area of Science:

  • Dynamical Systems Modeling
  • Scientific Machine Learning
  • Computational Science

Background:

  • Physics-informed machine learning (PIML) offers faster, more generalizable predictions for dynamical systems than classical methods.
  • Existing PIML models like RNNs, transformers, and neural operators struggle with long-time integration, long-range dependencies, chaotic dynamics, and extrapolation.
  • These limitations hinder the accurate and efficient prediction of complex dynamical systems.

Purpose of the Study:

  • Introduce state-space models implemented in Mamba for accurate and efficient dynamical system operator learning.
  • Address the limitations of current architectures in capturing long-range dependencies and computational efficiency.
  • Evaluate Mamba's performance against 11 baselines on strict extrapolation testbeds and a real-world application.

Main Methods:

  • Implemented state-space models using the Mamba architecture.
  • Developed novel extrapolation testbeds to rigorously evaluate model generalization.
  • Compared Mamba against 11 baseline models on interpolation and extrapolation tasks.
  • Applied Mamba to a quantitative systems pharmacology problem for drug efficacy assessment.

Main Results:

  • Mamba demonstrated superior performance in both interpolation and challenging extrapolation tasks.
  • Mamba consistently ranked among the top models with the lowest computational cost.
  • The model exhibited exceptional extrapolation capabilities, surpassing existing methods.
  • Mamba showed good performance in a real-world quantitative systems pharmacology application with limited data.

Conclusions:

  • Mamba presents a powerful and efficient tool for dynamical system operator learning.
  • Its ability to capture long-range dependencies and computational efficiency makes it suitable for complex scientific machine learning tasks.
  • Mamba shows significant potential for advancing research in dynamical systems modeling and quantitative systems pharmacology.