Which numerical methods exist to solve (estimate) completely arbitrary equations with:
- Constraints on definition and/or value space (boundary conditions for example)
- Composition with arbitrary smooth functions
- Differentiation and integration
- Probabilistic constraints.
For example
$$\exp\left[i\left(\frac{\partial f}{\partial x}\right)^2\right]= \sin(x^2)$$ Subject to $$\int_{y_{n}}^{y_{n+1}} f(x) dy = c_n$$
For prescribed $c_n$. The intent is in some sense a histogram integrating over slices of the value space.
(this integral may not make much sense right now, but I am working on it.)