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Forgive me for the poorly researched question. I'm currently working on a computer science project involving training a neural stochastic differential equation, and I've run into a problem while dealing with the loss.

Suppose I have an Ito process which is a solution to $$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t $$ Is there an estimate for $H(X_t)$ for fixed time $t$, in terms of $\mu$ and $\sigma$? I know about the Fokker-Planck equation but I have yet to figure out how I might use it to estimate $H(X_t)$.

Naively I could sample a bunch of $X_t$'s and bin them to obtain a kernel density estimate, from which I could approximate $H(X_t)$, but this is computationally inefficient in the context I'm working, and I wouldn't be able to propagate gradients through it.

Even a useful upper bound on $H(X_t)$ in terms of the drift and diffusion would be helpful. Any ideas or prior research someone can point me to? Thanks!

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  • $\begingroup$ Well,the entropy is just a particular integral over the transition density eg. $$E_{x_{0}}[g(X_{t})]=\int g(x) p_{t}(x_{0},x)dx$$, so yes a good approach is to approximate your particular Fokker-Plank. $\endgroup$
    – Thomas Kojar
    Commented Nov 19, 2023 at 22:07
  • $\begingroup$ @ThomasKojar Unfortunately I think our particular "Fokker-Plank" is about as general as it gets - our drift and diffusion are just neural networks, hence why I was curious if there are things we can say in general. $\endgroup$
    – user3002473
    Commented Nov 19, 2023 at 22:11
  • $\begingroup$ Not really. Most Fokker-Planks pdes don't even have any explicit solutions because they are highly nonlinear. However, Fokker-Plank pdes show up all over in physics, so there is a lot literature on doing simulations for it. Especially if you can get it in divergence-form. $\endgroup$
    – Thomas Kojar
    Commented Nov 19, 2023 at 22:14
  • $\begingroup$ If you have particular information on $\mu,\sigma$ eg. being some convolutions or some affine transformations, then we can try to get some bounds (put them in your post, not in the comments). But for the general setting, even uniqueness is not guaranteed eg. one often needs some Lipschitz/Hölder-regularity. $\endgroup$
    – Thomas Kojar
    Commented Nov 19, 2023 at 22:17
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    $\begingroup$ The literature is huge because it really depends on the coefficients. I would start looking on other neural-networks articles that took the SDE route (eg. this is common in stochastic gradient descent). Also NN themselves have been used to study FP equations eg. proceedings.mlr.press/v145/zhai22a/zhai22a.pdf or cims.nyu.edu/~wtu1/papers/2020Chaos.pdf "Neural network representation of the probability density function of diffusion processes" contains many references $\endgroup$
    – Thomas Kojar
    Commented Nov 19, 2023 at 22:30

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