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Related Concept Videos

Partial Differential Equations01:21

Partial Differential Equations

A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water...
Implicit Differentiation with Partial Derivatives01:27

Implicit Differentiation with Partial Derivatives

Implicit differentiation with partial derivatives is used when a relationship between two variables is defined implicitly rather than explicitly. Instead of solving one variable in terms of the other, the variables remain connected through a single equation. In this setting, one variable is treated as depending on the other, and differentiation is applied directly to the entire relation.To differentiate an implicit relation, the chain rule is applied to every term in the equation. Because one...
Implicit Differentiation: Problem Solving01:29

Implicit Differentiation: Problem Solving

Curves defined implicitly, where variables cannot be separated algebraically, require specialized techniques for analysis. The conchoid of Nicomedes exemplifies such a case. Its equation links x and y in a way that prevents isolation of one variable, making implicit differentiation essential to determine the slope and behavior at any point on the curve.The implicit form of the conchoid can be expressed as:To differentiate this equation, y is treated as a function of x, and the chain rule is...
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...
Differential Equations: Problem Solving01:21

Differential Equations: Problem Solving

When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
Introduction to Partial Derivatives01:25

Introduction to Partial Derivatives

In many real-world situations, an output depends on more than one input. In a high-tech assembly plant, total production may depend on technician labor and machine capacity at the same time. This relationship can be represented by a continuous function P(T, M), where T denotes technician labor input, and M denotes machine capacity. When demand increases, but the budget remains fixed, the manager must determine which input will improve production more efficiently.Partial derivatives provide a...

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Related Experiment Videos

Physics-informed neural networks for solving derivative-constrained partial differential equations.

Kentaro Hoshisashi1, Carolyn E Phelan1, Paolo Barucca1

  • 1University College London, Department of Computer Science, Gower Street, London WC1E 6BT, United Kingdom.

Physical Review. E
|June 19, 2026
PubMed
Summary
This summary is machine-generated.

Derivative-constrained PINNs (DC-PINNs) enhance physics-informed neural networks by incorporating derivative constraints for solving partial differential equations. This method improves physical fidelity and training stability for complex problems.

Related Experiment Videos

Area of Science:

  • Computational Science
  • Applied Mathematics
  • Machine Learning

Background:

  • Physics-informed neural networks (PINNs) solve PDEs via optimization but often miss fundamental derivative-based constraints.
  • Many real-world applications necessitate incorporating complex relationships involving states and their derivatives.

Purpose of the Study:

  • Introduce Derivative-Constrained PINNs (DC-PINNs) as a general framework for solving constrained PDEs.
  • Embed nonlinear derivative constraints efficiently using automatic differentiation.
  • Develop self-adaptive loss balancing to reduce hyperparameter tuning.

Main Methods:

  • Formulate constrained PDE solving as an optimization problem guided by a minimum objective function.
  • Incorporate general nonlinear constraints on states and derivatives (e.g., bounds, monotonicity, incompressibility).
  • Utilize automatic differentiation for efficient constraint computation and self-adaptive loss balancing for objective weighting.

Main Results:

  • DC-PINNs demonstrate consistent reduction in constraint violations and improved physical fidelity compared to baseline PINNs.
  • Successfully applied to benchmarks including heat diffusion with bounds, arbitrage-free financial volatilities, and vortex shedding in fluid flow.
  • Showcase improved training stability and convergence towards physically admissible solutions, even with small PDE residuals.

Conclusions:

  • DC-PINNs provide a robust and generalizable approach for solving constrained PDEs.
  • Explicitly encoding derivative constraints enhances the reliability and physical interpretability of solutions.
  • The framework offers a path towards more accurate and stable physics-informed machine learning models grounded in physical principles.