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Related Concept Videos

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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, the...
State Space Representation01:27

State Space Representation

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...
Classification of Systems-I01:26

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
Modeling with Differential Equations01:25

Modeling with Differential Equations

Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
Classification of Systems-II01:31

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
Mechanical Systems01:22

Mechanical Systems

Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass's acceleration. This interplay of forces is mathematically described...

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Related Experiment Video

Updated: Jul 2, 2026

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks
11:18

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks

Published on: March 2, 2015

Deep neural networks as discrete dynamical systems: Implications for physics-informed learning.

Abhisek Ganguly1, Santosh Ansumali1, Sauro Succi2,3,4,5

  • 1Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, Karnataka, India.

The Journal of Chemical Physics
|July 1, 2026
PubMed
Summary
This summary is machine-generated.

Physics-informed neural networks (PINNs) offer a novel computational approach for solving differential equations, interpreting deep neural networks (DNNs) as dynamical systems. This method provides flexibility, especially in high-dimensional problems.

Related Experiment Videos

Last Updated: Jul 2, 2026

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks
11:18

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks

Published on: March 2, 2015

Area of Science:

  • Computational Mathematics
  • Machine Learning
  • Dynamical Systems

Background:

  • Deep neural networks (DNNs) can be modeled as discrete dynamical systems.
  • Neural integral equations and partial differential equations (PDEs) share analogies with DNNs.
  • Physics-informed neural networks (PINNs) offer a novel approach to solving PDEs.

Purpose of the Study:

  • To compare numerical solutions of Burgers' and Eikonal equations with PINN-derived solutions.
  • To analyze DNNs as discrete dynamical systems and their computational pathways.
  • To evaluate the interpretability and computational cost of PINNs versus traditional methods.

Main Methods:

  • Comparative analysis of exact/numerical solutions and PINN solutions for Burgers' and Eikonal equations.
  • Interpreting DNNs as discrete dynamical systems approaching attractors.
  • Examining dense parameter representations learned by PINNs.

Main Results:

  • PINN learning presents a distinct computational pathway for approximating system dynamics.
  • DNNs function as discrete dynamical systems where layerwise evolution converges to attractors.
  • PINNs utilize dense, distributed parameter representations, unlike classical discretization stencils.

Conclusions:

  • PINNs offer a flexible alternative to classical numerical methods, particularly for high-dimensional problems.
  • The distributed representations in PINNs, while less interpretable and more computationally expensive, provide advantages in complex scenarios.
  • PINNs represent a significant advancement in solving differential equations by leveraging deep learning principles.