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Questions tagged [differential-equations]

A differential equation is an equation that contains at least one derivative.

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Does probability flow ODE trajectory (in the context of diffusion models) represents a bijective mapping between any distribution to a gaussian?

I have read several papers about diffusion models in the context of deep learning. especially this one As explained in the paper, by learning the score function (∇log(𝑝𝑡(𝑥))) ,probability flow ode ...
saleh's user avatar
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How can I coefficients for a model of n linear differential equations in n unknowns

I have (a lot of) time series data, to which I wish to fit a system of n linear differential equations in n unknowns, where by "fit" I mean to find coefficients on the unknowns in a way that ...
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Lotka-Volterra ordinary differential equation model to describe oscillations in a single observed entity

Background The Lotka-Volterra model is the starting point for any model of ecological dynamics. It can be described as a set of two ordinary differential equations (ODEs) with varying complexity. ...
Luka Seamus Wright's user avatar
8 votes
5 answers
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How can we efficiently find the fourth moment of a Poisson distribution?

Suppose we have $X\sim \textrm{Poisson}(\lambda)$ and we know that moment generating function $M(t)=\mathbb{E}(e^{tX})$. How do we use the moment generating function property $M^k(0)=\mathbb{E}(X^k)$ ...
Kai's user avatar
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Neural ODE and Adjoint method

I am trying to understand the paper NeuralODE which is very interesting. I get the general idea and the proof they give about the dynamics of the adjoint are fairly simple. They define a network $f$, ...
Julien Séveno-Piltant's user avatar
1 vote
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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 ...
HelloGoodbye's user avatar
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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 $ ...
Blademaster's user avatar
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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 ...
donut's user avatar
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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. \...
Looper's user avatar
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2 votes
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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 ...
jacob89's user avatar
1 vote
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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 ...
Learning_how_to_model's user avatar
2 votes
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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 ...
Loko Volperro's user avatar
10 votes
6 answers
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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{...
Joff's user avatar
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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|\...
user378967's user avatar
4 votes
2 answers
487 views

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 ...
Till Hoffmann's user avatar
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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 ...
Formal_that's user avatar
1 vote
0 answers
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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 ...
ForceBru's user avatar
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1 answer
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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, ...
S. Jeon's user avatar
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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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1 answer
134 views

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?
ironhide012's user avatar
1 vote
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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 ...
Marc Vaisband's user avatar
3 votes
1 answer
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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 ...
Joff's user avatar
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6 votes
2 answers
850 views

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 ...
foghorn's user avatar
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1 answer
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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 [...
Kell's user avatar
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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 ...
Christian Prince's user avatar
1 vote
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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 + \...
Kevin Choon Liang Yew's user avatar
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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 ...
krishnab's user avatar
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1 vote
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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. ...
mugdi's user avatar
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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 ...
krishnab's user avatar
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2 votes
1 answer
103 views

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 ...
donut's user avatar
  • 263
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1 answer
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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 ...
Paichu's user avatar
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3 votes
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98 views

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 $k$. ...
Thomas's user avatar
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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 ...
1 vote
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251 views

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 (...
auf-wiedersehen-yall's user avatar
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0 answers
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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 ...
student's user avatar
  • 261
2 votes
1 answer
229 views

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 ...
jbuddy_13's user avatar
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1 vote
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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 ...
Pedar's user avatar
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1 vote
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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 ...
Parinn's user avatar
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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 ...
jbuddy_13's user avatar
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1 vote
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39 views

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 ...
jeffery_the_wind's user avatar
0 votes
1 answer
436 views

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? ...
johnny_1010's user avatar
1 vote
0 answers
273 views

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. ...
Brian's user avatar
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8 votes
0 answers
267 views

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 ...
jeffery_the_wind's user avatar
9 votes
2 answers
3k views

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-...
Federico Poloni's user avatar
5 votes
1 answer
910 views

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) ...
Fraïssé's user avatar
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1 vote
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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 ...
yuno's user avatar
  • 11
8 votes
1 answer
1k views

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 ...
Edv Beq's user avatar
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