# 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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### 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? ...
1 vote
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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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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) ... • 1,071 1 vote 0 answers 31 views ### 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 ... • 11 7 votes 1 answer 600 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 ... • 721 0 votes 0 answers 43 views ### 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 ... 1 vote 0 answers 166 views ### 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. ... • 102 10 votes 1 answer 326 views ### 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 ... • 107 0 votes 0 answers 54 views ### 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 ... • 33.3k 0 votes 1 answer 200 views ### 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,... • 165 1 vote 1 answer 33 views ### 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 ... • 839 2 votes 0 answers 119 views ### 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 ... • 427 0 votes 1 answer 130 views ### 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 ... • 101 9 votes 2 answers 171 views ### 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 ... • 515 1 vote 0 answers 13 views ### 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} ... 0 votes 0 answers 81 views ### 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. ... • 33.3k 2 votes 2 answers 1k views ### 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 ... • 33.3k 2 votes 2 answers 388 views ### 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 ... • 33.3k 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 ... • 33.3k 2 votes 1 answer 1k views ### 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 ... • 31 3 votes 0 answers 171 views ### 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 ... 0 votes 0 answers 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 ... • 101 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: <... • 5,916 1 vote 0 answers 53 views ### 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 ... • 203 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 ... • 15.7k 0 votes 0 answers 49 views ### 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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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 ...