Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

60
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,...
60
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

85
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
85
Second Order systems II01:18

Second Order systems II

86
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
86
First Order Systems01:21

First Order Systems

83
First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
83
Linear time-invariant Systems01:23

Linear time-invariant Systems

209
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
209
Classification of Systems-I01:26

Classification of Systems-I

168
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:
168

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

VENI, VINDy, VICI: A generative reduced-order modeling framework with uncertainty quantification.

Neural networks : the official journal of the International Neural Network Society·2026
Same author

Reduced order modeling with shallow recurrent decoder networks.

Nature communications·2025
Same author

Unveiling the biological side of PET-derived biomarkers: a simulation-based approach applied to PDAC assessment.

European journal of nuclear medicine and molecular imaging·2024
Same author

Editor's Note: Low-Dose Paclitaxel Reduces S100A4 Nuclear Import to Inhibit Invasion and Hematogenous Metastasis of Cholangiocarcinoma.

Cancer research·2024
Same author

Uncertainty quantification and sensitivity analysis of neuron models with ion concentration dynamics.

PloS one·2024
Same author

Deep learning-based surrogate models for parametrized PDEs: Handling geometric variability through graph neural networks.

Chaos (Woodbury, N.Y.)·2023
Same journal

Anchor-based disentanglement framework for incremental multi-view clustering.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Complex-valued amplitude-phase interference modeling for adversarially robust classification.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

TraNce: Type-aware hypergraph neural network with biological mediators for drug repositioning.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Decentralized ADMM for factorization-based Low-rank matrix estimation.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Memristive neuromorphic circuit design inspired by the neural mechanisms of conditioned fear.

Neural networks : the official journal of the International Neural Network Society·2026
Same journal

Q-learning based asynchronous Boolean control networks stabilization with data loss.

Neural networks : the official journal of the International Neural Network Society·2026
See all related articles

Related Experiment Video

Updated: May 30, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

1.6K

On latent dynamics learning in nonlinear reduced order modeling.

Nicola Farenga1, Stefania Fresca1, Simone Brivio1

  • 1MOX, Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy.

Neural Networks : the Official Journal of the International Neural Network Society
|January 28, 2025
PubMed
Summary
This summary is machine-generated.

Latent Dynamics Models (LDMs) offer a novel mathematical framework for reduced order modeling of parameterized nonlinear time-dependent PDEs. This approach enhances accuracy and approximation capabilities for complex dynamical systems.

Keywords:
Approximation theoryDeep learningParameterized dynamical systemsReduced order modelingScientific machine learning

More Related Videos

Real-Time Proxy-Control of Re-Parameterized Peripheral Signals using a Close-Loop Interface
11:54

Real-Time Proxy-Control of Re-Parameterized Peripheral Signals using a Close-Loop Interface

Published on: May 8, 2021

4.3K
Quantifying Learning in Young Infants: Tracking Leg Actions During a Discovery-learning Task
11:18

Quantifying Learning in Young Infants: Tracking Leg Actions During a Discovery-learning Task

Published on: June 1, 2015

10.6K

Related Experiment Videos

Last Updated: May 30, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

1.6K
Real-Time Proxy-Control of Re-Parameterized Peripheral Signals using a Close-Loop Interface
11:54

Real-Time Proxy-Control of Re-Parameterized Peripheral Signals using a Close-Loop Interface

Published on: May 8, 2021

4.3K
Quantifying Learning in Young Infants: Tracking Leg Actions During a Discovery-learning Task
11:18

Quantifying Learning in Young Infants: Tracking Leg Actions During a Discovery-learning Task

Published on: June 1, 2015

10.6K

Area of Science:

  • Computational Mathematics
  • Scientific Machine Learning
  • Numerical Analysis

Background:

  • Reduced order modeling (ROM) is crucial for efficiently simulating parameterized nonlinear time-dependent partial differential equations (PDEs).
  • Existing methods often face challenges in accurately capturing the complex dynamics and parameter dependencies of these systems.
  • The need for robust and accurate approximation techniques for high-dimensional, time-dependent problems persists.

Purpose of the Study:

  • To introduce a novel mathematical framework, Latent Dynamics Models (LDMs), for the reduced order modeling of parameterized nonlinear time-dependent PDEs.
  • To develop a time-continuous and learnable approach that provides bounded approximation errors with respect to the full order model (FOM).
  • To enhance the interpretability and accuracy of reduced order models through a deep learning-based convolutional architecture.

Main Methods:

  • Formulation of LDMs as a nonlinear dimensionality reduction problem with a constrained latent dynamical system.
  • Derivation of error and stability estimates in a time-continuous setting.
  • Development of a time-discrete formulation (ΔLDM) using explicit Runge-Kutta schemes and a learnable variant (ΔLDMθ) employing deep neural networks (DNNs) and convolutional autoencoders with affine modulation.

Main Results:

  • The proposed LDM framework provides a time-continuous approximation of the FOM solution, enabling querying at arbitrary time instances with controlled accuracy.
  • The learnable ΔLDMθ formulation, utilizing DNNs, achieves bounded approximation errors relative to the FOM.
  • Numerical experiments with Burgers' and advection-diffusion-reaction equations validate the framework's accuracy and ability to handle parameterized PDEs.

Conclusions:

  • Latent Dynamics Models offer a mathematically rigorous and effective framework for reduced order modeling of complex, time-dependent parameterized PDEs.
  • The integration of deep learning, particularly convolutional architectures, enhances the spatial coherence and interpretability of latent representations.
  • The LDM approach significantly improves the accuracy and approximation capabilities of reduced order models, demonstrating broad applicability in scientific computing.