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Related Concept Videos

The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Woodward–Hoffmann Selection Rules and Microscopic Reversibility01:34

Woodward–Hoffmann Selection Rules and Microscopic Reversibility

Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
Constraints and Statical Determinacy01:26

Constraints and Statical Determinacy

In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
Collisions in Multiple Dimensions: Introduction01:05

Collisions in Multiple Dimensions: Introduction

It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a problem,...

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Temporal Ordering of Dynamic Expression Data from Detailed Spatial Expression Maps
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Temporal Ordering of Dynamic Expression Data from Detailed Spatial Expression Maps

Published on: February 9, 2017

From data chaos to physically interpretable deterministic mapping.

Dongni Jia1,2, Shuai Li1, Xinyi Zuo1,2

  • 1Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China.

Nature Communications
|July 13, 2026
PubMed
Summary

This study introduces a novel operator-learning framework for discovering governing equations from complex data. The method enhances stability and accuracy, even with noisy or shifted observations.

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High-Throughput Analysis of Optical Mapping Data Using ElectroMap
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High-Throughput Analysis of Optical Mapping Data Using ElectroMap

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Temporal Ordering of Dynamic Expression Data from Detailed Spatial Expression Maps
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Temporal Ordering of Dynamic Expression Data from Detailed Spatial Expression Maps

Published on: February 9, 2017

High-Throughput Analysis of Optical Mapping Data Using ElectroMap
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High-Throughput Analysis of Optical Mapping Data Using ElectroMap

Published on: June 4, 2019

Area of Science:

  • * Scientific computing and dynamical systems.
  • * Machine learning for scientific discovery.
  • * Data-driven modeling and equation inference.

Background:

  • * Discovering governing equations from observational data is challenging due to noise, high dimensionality, and multi-scale features.
  • * Traditional regression-based methods for equation discovery can suffer from limited structural stability and identifiability.
  • * Realistic data conditions often compromise the reliability of existing equation discovery techniques.

Purpose of the Study:

  • * To develop a robust framework for discovering governing equations from observational data.
  • * To overcome limitations of existing methods in handling noisy, high-dimensional, and multi-scale data.
  • * To achieve stable and interpretable equation recovery with long-horizon predictive accuracy.

Main Methods:

  • * A structured operator-learning framework integrating spectral decomposition, physics-guided sparse projection, and cross-view consistency regularization.
  • * Reformulation of equation discovery as a constrained dynamical inference problem.
  • * Decomposition of dynamics into scale-resolved components and enforcement of invariance across perturbed observations.

Main Results:

  • * Consistent identification of compact governing equations across various nonlinear systems and industrial processes.
  • * Strong long-horizon predictive accuracy maintained even with noisy and distribution-shifted data.
  • * Enhanced stability and interpretability in equation recovery compared to existing approaches.

Conclusions:

  • * The proposed framework offers a stable and reliable method for data-driven scientific discovery.
  • * It effectively addresses the challenges posed by realistic observational data in equation discovery.
  • * This approach advances the field of scientific machine learning for uncovering fundamental system dynamics.