Questions tagged [differential-equations]
A differential equation is an equation that contains at least one derivative.
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Efficiently sample from a limit set given a differential equation?
Given a dynamical system of many variables, described by an ordinary differential equation, is there some way to use machine learning to efficiently sample from the limit set (or maybe more accurately ...
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Fit birth-death ODE model incorporating uncertainty in measured data
I have the birth-death model $dy/dt = br \cdot x(t) - dr \cdot y(t)$ where the change of $y$ depends on the birth-rate $br$ and $x$ at timepoint $t$ and death rate $dr$ and $y$ at time point $t$.
$x(t)...
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Quantifying error in derived parameter from approximate solution to Laplace's equation
I am solving the Laplace equation $\nabla^2 u = 0$ on a 2D equispaced grid (spacing $ h $) with boundary conditions $ u(x=0) = 1 $ and $ u(x=L) = 0 $. My solver is approximate, yielding a solution $ ...
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Partial Differential Equation approach for ridgeplot
I have a series of probability density functions (pdfs) such that I can create a ridge plot with them evolving in time. Is it possible to know the intermediate function in between two of the functions ...
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Sampling from a Gaussian mixture (toy example ) using MALA
Say I want to sample from a Gaussian mixture $$\pi=\sum_{i=1}^3w_i\mathcal N(x_i,\sigma_i^2I_2)\tag1$$ where he support of the 3 distributions are "effectively separated"; e.g.
$w_1=.1$, $...
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Exact Successor State Distribution for a Pendulum
I want to solve the following problem. Suppose we have a simple pendulum, which follows the differential equation
\begin{equation}
\dot{x} = f(x) = [x_2, -\sin(x_1)]^T, \text{with } x=[x_1, x_2]^T.
\...
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Simulate SDE without error
Let
$d,k\in\mathbb N$;
$\sigma\in C^1(\mathbb R^d,\mathbb R^{d\times k})$ be Lipschitz and $\Sigma:=\sigma\sigma^\ast$;
$(W_t)_{t\ge0}$ be a $k$-dimensional Brownian motion;
$\lambda$ denote the ...
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Problem for a math formula in Weight Uncertainty in Neural Network
I am studying the paper "Weight Uncertainty in Neural Networks" by Blundell et al (2015, on arXiv), and there is a formula I don't get page 4, namely formula (3) in step 5:
I don't ...
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What's wrong with difference equations (disease modelling)?
I'm developing a mathematical (SIR-style) model for cholera transmission which I am fitting to data using MCMC. The model is reasonably complex (4 compartments, 6 parameters, simulation over 11200 ...
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Who formulated this Generalized Lotka-Volterra model? [closed]
In the introduction of this paper
Bifurcations in a predator–prey model with general logistic
growth and exponential fading memory
The authors claim
I didn't know that this model was well known.
Can ...
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How are differential equations related to Markov processes?
How can I classify the two terms differential equation and Markov process? Here are a few questions that I ask myself:
Is a Markov process a superset of differential equation or vice versa?
Do ...
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Why do Denoising Diffusion Probabilistic Models (DDPM) add noise according to $\sigma_t$ during sampling?
Reading about Denoising Diffusion Probabilistic Models (DDPM) the paper (algorithm 2 - sampling) states that the sampling goes according to...
$$
x_{t-1} = \frac{1}{\sqrt{\alpha_t}}\left( x_t - \frac{...
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Estimating parameters of differential equation from VAR model
I am reading section 3.7 of Algorithmic and high-frequency trading by Alvaro Cartea, Sebastian Jaimungal, and Josè Penalva. The book is not well written, with many typos - but nevertheless, it ...
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Is it possible to find an explicit equation for this maximum likelihood for a particular variable of mean function
I have a log-likelihood equation that involves multivariate normal. Let's say, $le = \sum_{i=1}^n logf(y_i)$ and
$f(y)=(2\pi)^{-\dfrac{n}{2}}|\sigma^2I_n|^{-\dfrac{1}{2}}exp[-\dfrac{1}{2}(y-x(t))^T|\...
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Gaussian process with Matérn kernel on a finite domain with periodic boundary conditions
I'm concerned with a Gaussian process $f(x)$ on a finite domain $x\in[0,L)$ with periodic boundary conditions. Naively using the distance after accounting for periodic boundary conditions to evaluate ...
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Estimating time varying parameters of ODE with the help of solution data
I am trying to extend a parameter estimation of ODE model from constant parameter estimation to time-varying parameter estimation. I have completed the constant parameter estimation (where parameter ...
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Are time-series models essentially difference equations with noise?
I've been studying some difference equations in my free time, and now I'm seeing them everywhere, especially in time-series models.
Here are some models that seem to be difference equations to me. I'm ...
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Tangent in specific point without model fitting
I have a question about differential function.
When I have one database with x and y, how can I find the slope in specific point?
For example,
...
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What are the deep learning books covering stochastic differential equations only?
I want to solve a simple Stochastic Differential Equation say $$dY=Y^2 dt+\sigma Y^2 dW$$ and then make future predictions.
I am conversant with MATLAB and LSTMs in python. Is there a book that can ...
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Explanation of Equation 5.80 in Pattern Recognition and Machine Learning - Bishop
How the equation 5.80 in _Pattern Recognition and Machine Learning_ by Bishop is derived?
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Memorylessness by way of additional dimensions
This is a somewhat broad question that occurred to me regarding the nature of memorylessness. Namely: Is there utility in considering systems which are themselves not memoryless, but then expanding ...
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How does the fixed point interation in invertible resnets work?
I feel like I am missing some easy point about this invertible resnet paper which is making it hard for me to grasp how the fixed point iteration works.
stated simply, the residual connection in a ...
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What is a "neural network prior" in this context of physics informed neural networks?
In the paper "Physics Informed Deep Learning (Part I): Data-driven solutions of nonlinear partial differential equations" (https://arxiv.org/abs/1711.10561v1), basically this paper uses a ...
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Validity of external cross validation using data generated by the fit model?
Context:
A paper I'm reading uses PDEs to characterise the effects of cancer treatments on the tumour microenvironment. The exact wording used in the paper is:
The predictive power of the [...
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How can one find a system of SDE's from a probability density function?
Suppose I have a joint distribution function say $p(x,y,z)=f_{X, Y, Z}(x,y,z)$. Is it possible to find a system of stochastic differential equations or a single stochastic differential equation from ...
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Partial derivative of a Group Lasso
I am looking at the gradient descent method for group lasso questions. Here's what I am currently stuck at.
Given the quadratic form of the objective function:
$$
f(x) = \frac{1}{2} x^T V x - m^T x + \...
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avoiding model overfitting when fitting parameters/models to an ordinary differential equation
I am working on fitting an ODE model to some data. So I have a vector of time series data $\textbf{x} = [x_1, x_2, ... x_n]$, and an ODE model $\dot{x} = f(x, \theta)$, where $\theta$ is a vector of ...
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Formal steps for gradient boosting with softmax and cross entropy loss function
Consider some data $\{(x_i,y_i)\}^n_{i=1}$ and a differentiable loss function $\mathcal{L}(y,F(x))$ and a multiclass classification problem which should be solved by a gradient boosting algorithm.
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Time-Series Analysis versus ODE parameter estimation -- why does one use first differences (stationarity) and the other does not?
This is kinda a tricky question, because it crosses disciplines. But I looking at the difference between time-series analysis in statistics, versus fitting parameters to ordinary differential ...
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1
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Find stochastic differential equation which best describes time-series
I have a time series with daily observations in a time span of 20 years describing the price of commodities. Given that this time series is non-stationary, is it possible to find a Stochastic ...
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What is the formula for AICc for least square fitting with multiple data types and variables?
If I have a system of nonlinear ordinary differential equations
\begin{align}
x' &= f(x,y,Q),\\
y' &= g(x,y,Q),
\end{align}
where $Q$ is the vector containing model parameters. And I fit it to ...
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Existence of the solution for SDE with Gaussian Process
I'm interested in the existence of the solution for a non-Ito SDE. Sloppy notation but assume a SDE given by
$\dot{x}=f(x),\quad f(x)∼GP(0,k(x,x′)),$
where $f$ is a Gaussian Process with kernel $k$. ...
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Recommnedation for books and material for solving differerntial equations through neural networks
I was going through some past and recent papers on using neural networks for solving ordinary and partial differential equations. One of the fascinating papers that inspired a flurry of papers is
...
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Does the Fisher information matrix have to be convex when assessing an optimal criteria? [closed]
I've reached a local minimum for a proposed model using a set of experimental data (positive definite Hessian) and want to select an additional experiment that will reduce the parameter uncertainties (...
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How to model stock price time series using differential equations?
I work with stock price time series where I check for structural breaks in the series. To do that I fit simple models such as AR and ARIMA.
However, I was proposed to express the stock price in terms ...
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Explanation of differential equation solution in survival analysis proof
I follow all the steps in the below derivation until the third to last line, "solving this differential equation for the survival analysis function shows that..."
Questions
I never took ...
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Find the correlation function of stochastic process given differential equations
Assume two systems for which the following differential equations hold between their input and output signals.
$$a \dfrac{dv(t)}{dt}+b v(t)=x(t)$$
$$\dfrac{dy(t)}{dt}=v(t)u(t)$$
Also, assume that the ...
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SIR: parameter estimation and optimization here (R)
From here https://ourworldindata.org/coronavirus/country/israel I have extracted the Covid Data for Israel, with some manipulations, I have obtained the plot of the daily new infections in Israel
If I ...
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How are differential equations and stochastic differential equations different?
In the simplest terms, how are differential equations and stochastic differential equations different?
As far as I can tell, SDEs are PDEs or ODEs, where the derivative of some function wrt itself is ...
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What does it mean to perform backprop "through the operations of an SDE solver"?
I am reading this cool paper about Neural SDEs as GANs. I've gotten through all of it and I understand fairly well. I've taken a couple classes on SDEs so I'm comfortable with the math. What I don't ...
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Deduce the Bellman equation from the Value and Q functions
I am trying to derive/deduce the bellman equation using Value and Q-functions.
I came only so far with understanding it and tried it myself in Latex:
Why is the $V^*$ suddenly in $Q^\pi$ function? ...
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R optim function - N parameters
I am working on SIR model. I try to estimate the parameters according to datas.
Here is a code similar to what I am working with.
...
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What's up with Neural Stochastic Differential Equations from a practical standpoint?
I've spent a few days reading some of the new papers about Neural SDEs. For example, here is one from Tzen and Raginsky and here is one that came out simultaneously by Peluchetti and Favaro. There are ...
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Why are most epidemic models continuous-time?
Most classical epidemic models such as SIR and variants are formulated as differential equations. However, to me discrete-time models feel more natural to measure the evolution of a disease on a day-...
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Is my understanding of neural ODE correct?
Given an ode $$\dot x = F(x(t))$$
The neural ODE model introduced in the paper: "Neural Ordinary Differential Equations" uses a neural network to model the solution of this ODE, i.e., $$x(t) ...
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Latent updates in Neural ODEs
I have read "Neural Ordinary Differential Equations" by Chen and coworkers and find it extremely interesting (https://arxiv.org/pdf/1806.07366.pdf).
There is one caveat that I seem to be ...
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On solving ode/pde with Neural Networks
Recently, I watched this video on YouTube on the solution of ode/pde with neural network and it motivated me to write a short code in Keras. Also, I believe the video is referencing this paper found ...
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Parameter estimation with differential equation?
I have an unknown function in an ODE as follows:
dx/dt=Q(t)-a*x(t),
a is a function of time and unknown. I plan to consider a as an unknown values at different time points. Then I use MCMC to estimate ...
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Adjoint relationship in Neural ODEs
The Chen et. al paper Neural ODE (https://arxiv.org/pdf/1806.07366.pdf) uses the adjoint method to take derivatives of solutions generated by an ODE solver with respect to neural network parameters θ. ...
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How does Hamiltonian Monte Carlo work?
I made the below graphic to explain how I currently understand the HMC algorithm. I'd like verification from a subject matter expert if this understanding is or isn't correct. The text in the below ...
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