Seminario de Probabilidad y Procesos Estocásticos, UNAM, 
Mayo 27,2026,
Saúl Diaz Infante Velasco

Modeling transition: ODE → SDE

https://slides.com/sauldiazinfantevelasco/seminario-proba-unam-2026

Two of the main approaches:

Numerical approximations

drift

diffusion

drift

diffusion

[Sec. 1.2, Thm 1.2.2]

Construction

References

State-of-the-Art SDE Software for SciML

Core Software Ecosystems

in SCi-ML

State-of-the-Art SDE Software for SciML

Software Package	Language	Key Solvers	SciML & Hardware Acceleration
DifferentialEquations.jl	Julia	Milstein, SRK (SRIW1, SOSRI), implicit methods.	The benchmark ecosystem. Features automatic differentiation, event handling, and automated GPU kernel generation. ^{[2, 3]}
torchsde	Python (PyTorch)	Euler-Maruyama, Reversible Heun, Milstein.	Native PyTorch tensor support. Specifically engineered for Neural SDEs, featuring highly scalable adjoint sensitivity methods.
Diffrax	Python (JAX)	Reversible Heun, implicit stochastic methods.	JAX-native. Exceptional for generating hardware-accelerated solvers for large ensembles of equations via vectorized mapping (vmap). ^[3]
Boost.odeint	C++	Runge-Kutta variants, Euler (customizable for SDEs).	Highly performant compiled backend. Supports CUDA and OpenCL directly, requiring C++ kernel development for peak performance. ^[3]
PyRates	Python (Code-Gen)	Interfaces with external libraries.	A code-generation tool that translates high-level network models into optimized backend implementations using Julia, PyTorch, or Fortran. ^[1]

References:

Gast, R., Knösche, T. R., & Kennedy, A. (2023). PyRates—A code-generation tool for modeling dynamical systems in biology and beyond. PLOS Computational Biology, 19, e1011761.
Rackauckas, C., & Nie, Q. (2017). DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. Journal of Open Research Software, 5, 15.
Utkarsh, U., et al. (2024). Automated translation and accelerated solving of differential equations on multiple GPU platforms. Computer Methods in Applied Mechanics and Engineering, 419, 116591.

State-of-the-Art SDE Software for SciML

Cybenko, G. Approximation by superpositions of a sigmoidal function. Math. Control, Signals Syst. 2, 303–314 (1989).

Solving With PINNs

\begin{aligned} \min_\theta & \frac{1}{N} \sum_{i=1}^N \Big \| \frac{d y_\theta}{d t}(t_i) - f(t_i, y_\theta(t_i)) \Big \| \\ & t_i \in [0, T] \end{aligned}

Becouse each solution to the differential equation is obtained by solving an optimization problem.

This framework has strong overtones of collocation methods or finite element methods. This is a popular line of work.

Zubov, K. et al. NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. arXiv (2021) doi:10.48550/arxiv.2107.09443.

Solving a SDE: classic numerics vs PINNs

Benchmark Example

Feature	Classic Methods	PINNs
Theory	Strong	Developing
Interpretability	High	Moderate
High Dimension	Hard	Better
Data Integration	Limited	Natural
Stability Guarantees	Yes	Often unclear

Neural ODEs and Neural SDEs.

Neural ODEs

A neural ordinary differential equation is a differential equation in which a neural network parameterizes the vector field.

The central idea now is to use a differential equation solver as part of a learned, differentiable computational graph (the sort of graph ubiquitous in deep learning).

\begin{aligned} \frac{d y}{d t}(t) & = f_\theta(t, y(t)) \\ & y(0) = y_0 \\ & f_\theta: \mathbb{R} \times \mathbb{R}^{d_1 \times \cdots \times d_k} \to \mathbb{R}^{d_1 \times \cdots \times d_k} \end{aligned}

Chen et al. Neural Ordinary Differential Equations. in Advances in Neural Information Processing Systems (eds. Bengio, S. et al.) vol. 31 (2018).