Stochastic partial differential equations in surface growth models

Stochastic partial differential equations (SPDEs) play a crucial role in modeling complex surface growth phenomena where randomness and noise are inherent aspects of the system. These equations extend traditional partial differential equations (PDEs) by incorporating stochastic processes, making them ideal for describing surface evolution under random influences like particle deposition, temperature fluctuations, or environmental noise. Below is a detailed exploration of SPDEs in surface growth models.

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sankey-beta
Code snippets,MATLAB,7
MATLAB,KPZ Model,1
MATLAB,EW Model,1
MATLAB,MH Model,1
MATLAB,noise in SPDEs,1
MATLAB,Finite Difference methods for SPDEs,1
MATLAB,Monte Carlo Simulations,1
MATLAB,Spectral Methods,1

Code snippets,Python,6
Python,KPZ Model,1
Python,EW Model,1
Python,MH Model,1
Python,noise in SPDEs,1
Python,Finite Difference methods for SPDEs,1
Python,Monte Carlo Simulations,1
Python,Spectral Methods,1

Code snippets,Julia,5
Julia,EW Model,1
Julia,MH Model,1
Julia,noise in SPDEs,1
Julia,Finite Difference methods for SPDEs,1
Julia,Monte Carlo Simulations,1

Code snippets,R,1
R,KPZ Model,1

Code snippets,C++,1
C++,KPZ Model,1

Code snippets to a typical SPDE used in surface growth

Code snippets to Edwards–Wilkinson (EW) Model

Code snippets to Mullins–Herring (MH) Model

Code snippets to Noise in SPDEs

Finite Difference Methods for SPDEs with code snippets

Monte Carlo Simulations for SPDEs with codes

Spectral Methods for SPDEs with codes

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🐳Ref.1

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Ref.,surface growth models,1
Ref.,Euclidean quantum field theories,1
Ref.,fluid dynamics,2
Ref.,stochastic evolution models of biological or chemical quantities,2
Ref.,interest-rate models,2
Ref.,stochastic filtering,3
Ref.,stochastic partial differential equations,9

Stochastic partial differential equations in Euclidean quantum field theories

Stochastic partial differential equations in fluid dynamics

Stochastic evolution models of biological or chemical quantities

Stochastic evolution models of interest-rate models

Stochastic evolution models of stochastic filtering

Stochastic partial differential equations

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🐳Ref.2

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Ref.,Galerkin methods,4
Ref.,physics-informed neural networks,1
Ref.,Fourier neural operators for parameteric PDEs,1
Ref.,deep neural networks,5
Ref.,neural SPDEs,1
Ref.,learning PDEs,3

Spatial discretizations based on Galerkin methods

Physics-informed neural networks to the solution of SPDEs

Fourier neural operators for parameteric PDEs

Fourier neural operators for deep neural networks

Fourier neural operators for neural SPDEs

The neural network applications for learning PDEs

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🐳Ref.3

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Ref.,the universal approximation property of neural networks,7
Ref.,random neural networks,6
Ref.,the Malliavin regularity,2
Ref.,the Stroock-Taylor formula,1
Ref.,the approximation rates for deterministic/random neural networks,2
Ref.,neural networks in the Wiener chaos expansion,10
Ref.,a system of coupled PDEs,5
Ref.,Fourier approaches,2
Ref.,Zakai equation in filtering,4
Ref.,the Adam algorithm,1

The universal approximation property of neural networks to obtain a universal approximation result for SPDEs

Random neural networks defined as single-hidden-layer neural networks whose weights and biases inside the activation function

The Malliavin regularity

The Stroock-Taylor formula

The approximation rates for deterministic/random neural networks

Learn the solution of (SPDE) with (possibly random)neural networks in the Wiener chaos expansion

A system of coupled PDEs

Zakai equation in filtering

The Adam algorithm

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🐳Ref.4

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Ref.,Jensen's inequality,1
Ref.,the vector-valued Stone-Weierstrass theorem,1
Ref.,Ito's isometry,1
Ref.,isonormal process,1
Ref.,the Wick polynomials,1
Ref.,the Stroock-Taylor formula,1
Ref.,Malliavin calculus for Hilbert space-valued random variables,4
Ref.,the chain rule for the Malliavin derivative,1
Ref.,the Malliavin derivative of a time integral,1
Ref.,the Malliavin derivative of a stochastic integral with F-predictable integrand ,1

Jensen’s inequality

The vector-valued Stone-Weierstrass theorem

Ito’s isometry

isonormal process

the Wick polynomials

the Stroock-Taylor formula

Malliavin calculus for Hilbert space-valued random variables

the chain rule for the Malliavin derivative

the Malliavin derivative of a time integral

the Malliavin derivative of a stochastic integral with F-predictable integrand

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