Questions tagged [differential-equations]

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

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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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ODE model selection criterion selection criteria

There is substantial literature on model selection criteria and many questions around this topic on CrossValidated. However, I could not find one that covers my case. I have a time series dataset (of ...
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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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Constant Mode Equation for a Weibull Distribution

I am trying to build a movement class for a simulated annealing algorithm for predicting an optimal spare parts policy. For better or worse I am looking to the Weibull distribution to move about the ...
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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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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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Differentiating a function with respect to a matrix [duplicate]

I'm new to matrix calculus and I'm trying to find the formulas for matrix differentiation. e.g. $\frac{\partial f}{\partial z}$ = zzx where z is a KxK matrix, and x is a vector in K I found a few ...
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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?

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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Modeling growth with repeated catalyst constributions

My data for one case looks like this Count P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10) P(11) P(12) P(13) P(14) P(15) time(t) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 catalyst(c) 1 2 3 4 1 ...
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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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When do moment equations reduce to the corresponding deterministic system?

If I have a stochastic system described by a Kolmogorov forward equation, I can derive the ODEs describing the moments. For example, $$ \frac{d\mathbb{E}[S]}{dt} = -\beta \mathbb{E}[SI] + g\mathbb{E}[...
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2 votes
1 answer
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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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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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5 votes
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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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3 votes
1 answer
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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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7 votes
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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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How can we modify SIR model to account multiple waves of infection?

We had a very nice discussion for modeling covid19 data with SIR model. If we monitor number of infected cases over time, most of these models will only have one wave. How can we modify the model to ...
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1 answer
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Neural ODEs gradient calculation for multiple time steps

I was reading the paper on Neural ODEs (here) and was wondering if anyone could offer some insight on calculation of the gradient of the loss function. If we are only considering 2 time points, $t_0,...
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1 vote
1 answer
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Why "run the filter longer than needed and remove the initial values" will solve the issue of recursive solving equations?

Consider sequence of random variables $w_i$ iid normal(0,1). Given the equation, $x_t=x_{t-1}-0.9x_{t-2}+w_t$ with $t$ discrete, I want to solve for $x_t$ recursively by prescribing $x_1,x_2$. The ...
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2 votes
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Differential Equation Prediction Model for Future Revenue

I'm trying to predict time series with the physical model of the process. Simple heuristic model for the predicting the company's future revenue. The hypothesis for the model are: The company's ...
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0 votes
1 answer
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Fixing convergence in SIR model using modified fit-model to fit COVID-19 data

I'm trying to model the data for covid-19 using SIR model in R. I followed the answer of the question, and the blog. I'm using the suggested code, However, the data does not converging. Any suggestion ...
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9 votes
2 answers
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Does this interpretation $\phi'(x)=-x\phi(x)$ of the normal distribution have any significance?

For the standard normal distribution $\phi(x)$, we can see that $\phi'(x)=-x\phi(x)$. Put differently, $\frac{\mathrm{d}\ln(\phi(x))}{\mathrm{d} x}= -x $. I see this as the fall in the value of the ...
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eigenstructure matching optimization

Is there any optimization loss functions that can approximately match the eigenstructure of the original samples and the transformed samples? For example, given a collection of samples $\mathbf{X}$ ...
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Fitting covid19 data with SIR model: question about the definitions of susceptible and recovered population in real world

We have some very nice discussions about SIR model fitting at CV. As I explore the model with different parameters, I have some questions on the definitions of susceptible and recovered population. ...
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2 votes
2 answers
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What are the pros and cons to fit data with simple polynomial regression vs. complicated ODE model?

Suppose in a disease outbreak scenario and we want to estimate number of infected people based infections over time. Why we cannot simply fit the data with some polynomials (or some MLP neural ...
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2 votes
2 answers
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Are the parameters $\beta$ and $\gamma$ in (Susceptible, Infected, Recovered) SIR model probability number? Can they larger than 1.0?

I am learning SIR model from this blog post. We also had a very good discussion in CV post The key parameters of the model are $\beta$ and $\gamma$, people usually describe them as the "infection ...
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9 votes
4 answers
410 views

Where does the logistic function come from?

I first learned the logistic function in machine learning, where it is just a function that map a real number to 0 to 1. We can use calculus to get the derivative and use it for some optimization ...
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2 votes
1 answer
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Neural ODE's, Adjoint Method

I've been trying to understand the gist behind the Chen et. al paper on neural ODE's (https://arxiv.org/pdf/1806.07366.pdf). It seems like the main trick here is to be able to take derivatives of ...
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3 votes
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Uncertainty propagation in ODEs

I want to see the effect of parameter uncertainty in the Euler method for ODEs. For a differential equation: $dx/dt=f$ with initial condition $x(0)=xo$ and a function $f$ (that has uncertain ...
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313 views

Can neural ODEs "fit" an ODE from just measurements?

The neural ODE technique, to my knowledge, presents a neural network based way of solving ODEs efficiently, which implies it needs an ODE and an initial value in order to construct the evolution over ...
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11 votes
5 answers
7k views

Fitting SIR model with 2019-nCoV data doesn't conververge

I am trying to calculate the basic reproduction number $R_0$ of the new 2019-nCoV virus by fitting a SIR model to the current data. My code is based on https://arxiv.org/pdf/1605.01931.pdf, p. 11ff: <...
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1 vote
0 answers
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Correct way to write a stochastic differential equation

A few days ago, I disccussed my Phd thesis (thesis defense), one of the mathematical mistakes that the committe members alerted me to it, is how to write correctley an ordinary differential equation ...
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22 votes
1 answer
10k views

What are the practical uses of Neural ODEs?

"Neural Ordinary Differential Equations", by Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt and David Duvenaud, was awarded the best-paper award in NeurIPS in 2018 There, authors propose the ...
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How to show CDF uniquely determined from equation

I have the following equation for a CDF that I would like to show is uniquely determined: $$\frac{h_1(x)}{f(x)}=\int_x^a h_2(z)(F(z)-F(x))^k\;dF(z)$$ Here $F$ is the CDF and $f$ is the PDF, which I ...
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2 votes
1 answer
5k views

derivative of mathematical expectation [closed]

As we know,if x is a random variable, we could write mathematical expectation based on cumulative distribution function $(F)$ as follow: $E(X)=\int[1-F(x)]d(x) $ In my problem, t is a random variable ...
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