Seminario de Ciencia de Datos, UAM Iztapalapa
Junio 18,2026,
Saúl Diaz Infante Velasco saul.diazinfante@unison.mx

Aprendizaje automático para dinámicas estocásticas en bioprocesos:

PINNs y SNODEs aplicados al crecimiento logístico en la producción de Kéfir.

https://slides.com/sauldiazinfantevelasco/seminario-datascience-uam-2026-06-18

https://slides.com/d/foDYQBY/live

01

Modeling Transition: From ODE to SDE

02

Classical Numerical Approaches

03

Scientific Machine Learning (SciML)

04

Neural ODEs and SDEs

A journey through the integration of physical models and machine learning for stochastic bioprocesses.

05

Application: Bioprocess Case Study

Modeling transition:

ODE → SDE

Two of the main approaches:

Numerical approximations for SDEs

drift

diffusion

Taylor explicit methods

drift

diffusion

[Sec. 1.2, Thm 1.2.2]

Discrete Brownian motion

Construction

References

Numerical Solvers for Stochastic Differential Equations

Classical simulation packages, statistical SDE toolboxes, differentiable solvers, and compiled/HPC workflows

Package	Ecosystem	Solver focus	Best use in an SDE exposition
sde Stefano M. Iacus classic	R	Euler, Milstein, KPS, Milstein2, conditional density, Ozaki, Shoji, and exact-algorithm simulation.	Natural reference package for simulation and inference of diffusion processes in the statistical SDE tradition. [R1] [R2]
YUIMA Simulation and inference for stochastic processes inference	R	SDEs driven by Wiener processes, Lévy processes, fractional Brownian motion, plus CARMA, COGARCH, and point-process models.	Best for statistically oriented SDE workflows: model specification, simulation, quasi-likelihood inference, hypothesis testing, change-point analysis, and high-frequency data methods. [R10]
Sim.DiffProc Simulation of diffusion processes	R	Euler-Maruyama, Milstein, predictor-corrector, Itô-Taylor, Heun, and stochastic Runge-Kutta schemes.	Good for a methods-oriented slide because the package makes the numerical hierarchy of strong and weak schemes explicit. [R3]
MATLAB `sde` Financial Toolbox Picchini SDE Toolbox	MATLAB	Vector SDE models, drift/diffusion objects, Brownian risk sources, and Euler-type sample-path simulation.	MATLAB analogue for finance-oriented SDE simulation; the Picchini toolbox is useful as a historical simulation/inference reference. [R4] [R5]
sdeint NumPy/SciPy-style interface pedagogical	Python	Itô and Stratonovich integration: Euler-Maruyama, Heun, SRI2/SRS2, and Kloeden-Platen implicit methods.	Useful Python baseline for teaching classical schemes before moving to neural or differentiable SDE solvers. [R6]
torchsde Differentiable SDE solvers research	PyTorch	GPU-compatible SDE solvers, stochastic adjoint sensitivity, and neural SDE training workflows.	Best positioned when the exposition includes neural SDEs, backpropagation through stochastic dynamics, or learned drift/diffusion. [R7] [R8]
StochasticDiffEq.jl DifferentialEquations.jl ecosystem broadest	Julia	EM, Heun, SOSRI/SOSRA, weak-order methods, stiff solvers, commutative-noise methods, jump-diffusion, GPU ensembles.	Strong general-purpose open-source SDE solver ecosystem; ideal when solver breadth, adaptivity, callbacks, and performance matter. [R9]
XMDS2 C++ code generation HPC / SPDE	C/C++	High-level model specification with generated, parallelized C++ code for stochastic systems.	Best presented as the compiled/HPC route rather than as a direct one-to-one analogue of Iacus’s scalar diffusion package. [R11]

Suggested reading order: statistical SDE packages → MATLAB/Python baselines → differentiable solvers → Julia/HPC ecosystems.

References

Classical SDE simulation packages and MATLAB/Python baselines

S. M. Iacus. Simulation and Inference for Stochastic Differential Equations: With R Examples. Springer, 2008.

S. M. Iacus. sde.sim: Simulation of stochastic differential equations. R package sde, CRAN documentation.

A. C. Guidoum and K. Boukhetala. Performing parallel Monte Carlo and moment equations methods for Itô and Stratonovich stochastic differential systems: R package Sim.DiffProc. Journal of Statistical Software, 96(2):1–82, 2020.

MathWorks. sde: Stochastic differential equation model. Financial Toolbox documentation.

U. Picchini. SDE Toolbox: Simulation and estimation of stochastic differential equations with MATLAB. MATLAB toolbox, 2007.

M. J. Aburn. sdeint: Numerical integration of stochastic differential equations. Python package documentation.

References

Differentiable SDE solvers, SciML ecosystems, YUIMA, HPC, and numerical SDE foundations

X. Li, T.-K. L. Wong, R. T. Q. Chen, and D. Duvenaud. Scalable gradients for stochastic differential equations. In Proceedings of AISTATS, PMLR 108:3870–3882, 2020.

P. Kidger, J. Foster, X. Li, and T. Lyons. Efficient and accurate gradients for neural SDEs. Advances in Neural Information Processing Systems, 34, 2021.

C. Rackauckas and Q. Nie. DifferentialEquations.jl: A performant and feature-rich ecosystem for solving differential equations in Julia. Journal of Open Research Software, 5:15, 2017.

R10

S. M. Iacus and N. Yoshida. Simulation and Inference for Stochastic Processes with YUIMA: A Comprehensive R Framework for SDEs and Other Stochastic Processes. Springer, Use R! series, 2018. doi:10.1007/978-3-319-55569-0.

R11

G. R. Dennis, J. J. Hope, and M. T. Johnsson. XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations. Computer Physics Communications, 184:201–208, 2013.

R12

P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, 1992.

Scientific Machine Learning (SciML)

State-of-the-Art SDE Software for SciML

Numerical solvers, neural SDEs, GPU acceleration, and model-code-generation tools

Software	Stack	Numerical focus	SciML role and references
StochasticDiffEq.jl DifferentialEquations.jl ecosystem	Julia	EM, Heun, SOSRI/SOSRA, weak-order, stiff, and jump-diffusion methods.	Strong general-purpose SDE baseline for SciML; integrates with callbacks, AD workflows, events, and GPU ensembles. [R1] [R2]
torchsde Differentiable SDE solvers archived	PyTorch	Neural SDEs, adjoint sensitivities, Brownian reconstruction, reversible Heun.	Natural reference for neural SDE experiments and gradient-based learning of stochastic dynamics. [R3] [R4]
Diffrax JAX-native solver suite	JAX	Unified ODE/SDE/CDE solvers, adjoints, PyTrees, and vmappable workflows.	Good fit for differentiable simulation, neural differential equations, and hardware-accelerated JAX pipelines. [R5]
Boost.odeint Compiled numerical backend ODE core	C++	Generic ODE steppers; useful for custom low-level stochastic time stepping.	Valuable when compiled control, C++ integration, or custom kernels matter; not an SDE-specialized SciML stack. [R6]
PyRates Model-code generation	Python	High-level model specification with backend code generation.	Useful for biological dynamical systems and reproducible model pipelines; better viewed as code generation than as a direct SDE solver. [R7]

Click software names for documentation or repositories. Click reference tags for the corresponding paper or source.

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 |^2 + \lambda |y_{\theta}(0) - y_0| \\ & 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.

Let

$$\frac{dy_{\theta}}{dt} = f(t, y_{\theta}(t)), \quad t \in (0, T],$$

with a known initial condition:

$$y_{\theta}(0) = y_0.$$

So, we want an NN $ y_{\theta}$ with parameters $\theta$ that satisfies the above equation.

For tuning a given NN architecture, we define the Loss function

$$ L(\theta) = L_{\text{residual}} + \lambda L_{\text{IC}}.$$

We choose a set of collocation points,

$\{t_i\}_{i=1}^{N}$ , spread across the interval ( $0, T]$ . At each point, we compute the residual:

$$\text{Residual}(t_i) = \frac{dy_\theta}{dt}(t_i) - f(t_i, y_\theta(t_i)).$$

$$L_{\text{IC}} = |y_\theta(0) - y_0|^2$$

L_{\text{residual}} = \frac{1}{N} \sum_{i=1}^N \Big | \frac{d y_\theta}{d t}(t_i) - f(t_i, y_\theta(t_i)) \Big |^2

The Challenge: Simulating SDEs

Consider a general scalar SDE,

$$dX_t = f(X_t) dt + g(X_t) dW_t$$

Fokker-Planck equation (Kolmogorov forward):

$$\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x} [f(x) p(x,t)] + \frac{1}{2} \frac{\partial^2}{\partial x^2} [g^2(x) p(x,t)]$$

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

References

D. J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3):525–546, 2001. doi:10.1137/S0036144500378302
M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019. doi:10.1016/j.jcp.2018.10.045
C.-W. Kong, L. Laurenti, J. McMahon, and M. Lahijanian. Error bounds for physics-informed neural networks in Fokker-Planck PDEs. In Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, Proceedings of Machine Learning Research, volume 286, pages 2291–2324, 2025. PMLR source

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).

Minimizing the loss function can be achieved using gradient-based methods, then we need an effective approach to computing the gradient.

The adjoint equation from optimal control theory is one of the major tools.

The ﬁrst derivation of the adjoint equation goes back to the idea of using Lagrangian formalism, where λ, the variable in the adjoint equation, plays a similar role as a Lagrange multiplier

Results that solving the adjoint equation backward in time provides the gradient of the loss.

\begin{aligned} \frac{d a_y}{d t}(t) &= -a_y(t)^\top \frac{\partial f_\theta}{\partial y}(t, y(t)), \\ & a_y(T) = \frac{d L}{d y(T)}, \\ \frac{d a_\theta}{d t}(t) & = -a_y(t)^\top \frac{\partial f_\theta}{\partial \theta}(t, y(t)). \\ & a_\theta(T) = 0, \end{aligned}

$y_0 \in \mathbb{R}^d$ and $ \theta \in \mathbb{R}^m $.

be continuous in $t$, uniformly Lipschitz in $y$, and continuously differentiable in $y$.

Let

be the unique solution to

\[y(0) = y_0, \qquad \frac{d y}{d t}(t) = f_\theta(t, y(t))\]

Let$L = L(y(T))$ be function of the terminal value $y(T)$.

Then $\frac{d L}{d y(t)} = a_y(t)$ and $\frac{d L}{d \theta} = a_\theta(0)$,

where

$a_y : [0, T] \to \mathbb{R}^d$

solve

f_\theta : [0, T] \times \mathbb{R}^d \to \mathbb{R}^d

y_{\theta} : [0, T ] \to \mathbb{R}^d

a_\theta : [0, T] \to \mathbb{R}^m

In sequence modeling and time-series analysis, the loss function is typically is a function of the state at multiple discrete time snapshots

The optimization problem then becomes:

Adjoint Method for Multi-Label Loss Functions

\{t_1, t_2, \dots, t_N\}

\begin{aligned} &\min_{\theta} L(z(t_1), z(t_2), \dots, z(t_N)) \\ \text{s.t.} & \\ &\dot{z}(t) = f(t, z(t), \theta) \end{aligned}

$$\left( \frac{\partial L}{\partial \theta} \right)_k = \sum_{i=1}^{N} \frac{\partial L}{\partial z(t_i)} \frac{\partial z(t_i)}{\partial \theta_k}$$

$$\frac{\partial z(t_i)}{\partial \theta_k} = \int_{t_0}^{t_N} \frac{\partial z(t)}{\partial \theta_k} \delta(t - t_i) dt$$

$$\left( \frac{\partial L}{\partial \theta} \right)_k = \int_{t_0}^{t_N} \frac{\partial z(t)}{\partial \theta_k} \left( \sum_{i=1}^{N} \frac{\partial L}{\partial z(t_i)} \delta(t - t_i) \right) dt$$

$$\left( \frac{\partial L}{\partial \theta} \right)_k = \left\langle \sum_{i=1}^{N} \frac{\partial L}{\partial z(t_i)} \delta(t - t_i), \frac{\partial z(t)}{\partial \theta_k} \right\rangle_{X^*, X}$$

\left( \frac{\partial L}{\partial \theta} \right)_k = \left\langle \frac{\partial L}{\partial z} , \frac{\partial z(t)}{\partial \theta_k} \right\rangle_{X^*, X}

\begin{aligned} &\min_{\theta} L(z(t_1), z(t_2), \dots, z(t_N)) \\ \text{s.t.} & \\ &\dot{z}(t) = f(t, z(t), \theta) \end{aligned}

By differentiating both sides of the restriction equation respect to $\theta$

\frac{d}{d\theta} \dot{z} = \frac{\partial f}{\partial z} \frac{\partial z}{\partial \theta} + \frac{\partial f}{\partial \theta}

\Big( \frac{d}{dt} - \frac{\partial f}{\partial z} \Big) \frac{\partial z}{\partial \theta} = \frac{\partial f}{\partial \theta}

Suppouse $\lambda$ = $\lambda(t)$ s.t.

\begin{aligned} \left\langle \frac{\partial L}{\partial z} , \frac{\partial z(t)}{\partial \theta_k} \right\rangle_{X^*, X} =& \left\langle \lambda , \Big( \frac{d}{dt} - \frac{\partial f}{\partial z} \Big) \frac{\partial z(t)}{\partial \theta_k} \right\rangle \\ =& \left\langle \lambda , \frac{\partial f}{\partial \theta_k} \right\rangle \\ =& \left\langle \Big( -\frac{d}{dt} - \frac{\partial f}{\partial z} \Big)^{\top} \lambda , \frac{\partial z}{\partial \theta_k} \right\rangle \end{aligned}

Integrating by parts

\begin{aligned} -\frac{d}{dt} \lambda^{\top}(t) &= \sum_{i=1}^N \frac{\partial L}{\partial z(t_i)} \delta(t-t_i) + \lambda^{\top} \frac{\partial f}{\partial z} \\ &\lambda(t_N) = 0 \\ -\frac{d}{dt} \lambda^{\top}(t) & = \lambda \frac{\partial f}{\partial z} \\ &\lambda^{\top}(t_i^{-}) = \lambda(t_i^{+}) + \frac{\partial L}{\partial z(t_i)} \\ & i = N-1, \dots, 1 \end{aligned}

\Big( \frac{\partial L}{\partial \theta} \Big)_k = \int_{t_N}^{t_0} \lambda(t)^{T} \frac{\partial}{\partial \theta_k} f(t, \textcolor{green}{z(t,\theta)}, \theta)

\begin{aligned} &\min_{\theta} L(z(t_1), z(t_2), \dots, z(t_N)) \\ \text{s.t.} & \\ &\dot{z}(t) = f(t, z(t), \theta) \end{aligned}

dZ_t = b_\theta(Z_t,t)\,dt+\sigma_\theta(Z_t,t)\,dW_t, \qquad Z_0=z_0

Nerural-SDE

We simulate the SDE forward, obtain $Z_T$ and compute the loss $ZTZ_T, compute a loss$

$L=ℓ(ZT),\mathcal L = \ell(Z_T)$

\mathcal L = \ell(Z_T)

However, the backward pass must use the same Brownian path as the forward pass.

Naively reversing an Itô SDE is not valid. The clean reverse-time construction uses the Stratonovich form:

dZ_t = b_\theta(Z_t,t)\,dt + \sum_{i=1}^m \sigma_{\theta,i}(Z_t,t)\circ dW_t^{(i)}

For a fixed Brownian sample path, the Stratonovich SDE defines a stochastic flow

Z_t = \Phi_{s,t}(Z_s)

Z_T = Z_0 + \int_0^T b_\theta(Z_t,t)\,dt + \sum_{i=1}^m \int_0^T \sigma_{\theta,i}(Z_t,t)\circ dW_t^{(i)}

Backward reconstruction of the state

The inverse stochastic flow reconstructs the earlier state from the terminal state:

Z_s = Z_T - \int_s^T b_\theta(Z_r,r)\,dr - \sum_{i=1}^m \int_s^T \sigma_{\theta,i}(Z_r,r)\circ d\widehat W_r^{(i)}

a_t = \frac{\partial \mathcal L}{\partial Z_t}

a_T = \frac{\partial \ell}{\partial Z_T}

a_0 = \frac{\partial \mathcal L}{\partial Z_0}, \qquad \frac{\partial \mathcal L}{\partial \theta}.

a_s = a_T + \int_s^T a_r \frac{\partial b_\theta}{\partial z}(Z_r,r)\,dr + \sum_{i=1}^m \int_s^T a_r \frac{\partial \sigma_{\theta,i}}{\partial z}(Z_r,r) \circ d\widehat W_r^{(i)}

da_t = a_t \frac{\partial b_\theta}{\partial z}(Z_t,t)\,dt + \sum_{i=1}^m a_t \frac{\partial \sigma_{\theta,i}}{\partial z}(Z_t,t) \circ d\widehat W_t^{(i)}

a_T = \frac{\partial \ell}{\partial Z_T}

q_t = \frac{\partial \mathcal L}{\partial \theta} \quad q_T = 0

q_s = q_T + \int_s^T a_r \frac{\partial b_\theta}{\partial \theta}(Z_r,r)\,dr + \sum_{i=1}^m \int_s^T a_r \frac{\partial \sigma_{\theta,i}}{\partial \theta}(Z_r,r) \circ d\widehat W_r^{(i)}

\frac{\partial \mathcal L}{\partial \theta} = q_0.

References: Physics-Informed Neural Networks & UQ

Z. Zou, X. Meng, A. F. Psaros, and G. E. Karniadakis. NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators. SIAM Review, 66(1):161–190, 2024. doi:10.1137/21M1446859
A. F. Psaros, X. Meng, Z. Zou, L. Guo, and G. E. Karniadakis. Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons. Journal of Computational Physics, 477:111902, 2023. doi:10.1016/j.jcp.2022.111902
M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019. doi:10.1016/j.jcp.2018.10.045

References: Neural ODEs & SDEs

R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31, 2018. NeurIPS source
E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equations I: Nonstiff problems. Springer Series in Computational Mathematics, volume 8, second edition, Springer, 1993. doi:10.1007/978-3-540-78862-1
P. Kidger, R. T. Q. Chen, and T. J. Lyons. “Hey, that’s not an ODE”: Faster ODE adjoints via seminorms. In Proceedings of the 38th International Conference on Machine Learning, Proceedings of Machine Learning Research, 139:5443–5452, 2021. PMLR source
X. Li, T. K. L. Wong, R. T. Q. Chen, and D. K. Duvenaud. Scalable gradients for stochastic differential equations. In Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), Proceedings of Machine Learning Research, 108:3870-3882, 2020. PMLR source
P. Kidger, J. Foster, X. Li, and T. J. Lyons. Neural SDEs as Infinite-Dimensional GANs. In Proceedings of the 38th International Conference on Machine Learning, Proceedings of Machine Learning Research, 139:5453-5463, 2021. PMLR source

Fitting the data of water kefir production

Example with real data

Kefir Water production

Dr. José Carlos Rodríguez Figueroa

The right-hand side $f_\theta$ is a fully connected neural network with input
$ (\tau,z) \in \mathbb R^2$, two hidden layers, 32 hidden units per layer, and
hyperbolic tangent activation:

\frac{d z}{d\tau} = f_\theta(\tau,z), \qquad z(0)=z_0.

\begin{aligned} h_1 &= \tanh(W_1 x+b_1), \\ h_{\ell} &= \tanh(W_{\ell}h_{\ell-1}+b_{\ell}), \\ f_\theta(x) &= W_o h_2+b_o \end{aligned}

\mathcal L_{\mathrm{ODE}}(\theta) = \frac{1}{n} \sum_{(i,j)\in\Omega} \left(z_i^{[j]}-\widehat z_i(\theta)\right)^2

\begin{aligned} \hbox{input to hidden 1:}&\quad 2\cdot 32+32=96,\\ \hbox{hidden 1 to hidden 2:}&\quad 32\cdot 32+32=1056,\\ \hbox{hidden 2 to output:}&\quad 32\cdot 1+1=33. \end{aligned}

\frac{du_\theta}{d\tau} = (t_{\max}-t_{\min})\, r u_\theta(\tau)\left(1-\frac{u_\theta(\tau)}{K}\right).

Let $\{\xi_m\}_{m=1}^M$ be $M=128$ uniformly spaced collocation points in $[0,1]$.

\mathcal L_{\mathrm{data}}(\theta) = \frac{1}{n} \sum_{(i,j)\in\Omega} \left( \frac{u_\theta(\tau_i)-y_i^{[j]}}{s_y} \right)^2,

\mathcal L_{\mathrm{init}}(\theta) = \frac{1}{J} \sum_{j=1}^J \left( \frac{u_\theta(0)-y_0^{[j]}}{s_y} \right)^2,

\mathcal L_{\mathrm{phys}}(\theta,r,K) = \frac{1}{M} \sum_{m=1}^M \left[ \frac{ \dfrac{du_\theta}{d\tau}(\xi_m) - (t_{\max}-t_{\min})r u_\theta(\xi_m) \left(1-\dfrac{u_\theta(\xi_m)}{K}\right) }{s_y} \right]^2

The deterministic PINN

\mathcal L_{\mathrm{detPINN}} = \mathcal L_{\mathrm{data}} + \lambda_0\mathcal L_{\mathrm{init}} + \lambda_{\mathrm{phys}}\mathcal L_{\mathrm{phys}},

NODE fit

\mathrm{SSE} = \sum_{(i,j)\in\Omega}(y_i^{[j]}-\widehat y_i^{[j]})^2, \qquad \mathrm{SST} = \sum_{(i,j)\in\Omega}(y_i^{[j]}-\bar y)^2.

\mathrm{RMSE} = \sqrt{\frac{\mathrm{SSE}}{n}},

R^2 = 1-\frac{\mathrm{SSE}}{\mathrm{SST}}

\widehat\sigma^2 = \frac{\mathrm{SSE}}{n}

\log \widehat L = -\frac{n}{2} \left[ \log(2\pi)+1+\log(\widehat\sigma^2) \right]

\mathrm{AIC}=2k-2\log\widehat L, \qquad \mathrm{BIC}=k\log(n)-2\log\widehat L

Evaluation Criteria

Water Kefir Model Comparison

Original response scale. Lower RMSE/AIC/BIC is better; higher R² is better.

Model	RMSE ↓	R² ↑	AIC ↓	BIC ↓	p	Main reading
Classical logistic ODE	1.4291	0.9826	165.84	171.26	2	Most parsimonious
Neural ODE	1.0281	0.9910	2502.20	4644.90	1185	Best in-sample fit
Neural SDE	2.0312	0.9649	4933.48	9217.08	2370	Uncertainty model
Deterministic logistic PINN	2.8167	0.9326	4644.90	8641.24	2211	Physics-constrained fit
Stochastic logistic PINN drift	2.8469	0.9311	4647.87	8646.01	2212	Logistic drift + noise estimate
Stochastic logistic SDE mean	3.2324	0.9112	4659.30	8657.44	2212	Mean of fitted stochastic process

Takeaway: Neural ODE gives the best interpolation accuracy, while the classical logistic ODE is preferred by AIC/BIC because it uses only two fitted dynamical parameters.

Stochastic Dynamics and PINN Parameters

Stochastic models quantify uncertainty; PINNs recover interpretable logistic parameters.

Stochastic-models
Model	90% coverage ↑	Mean width ↓
Neural SDE	0.9333	7.3887
Stochastic logistic SDE	0.8222	9.4517

Neural SDE gives better empirical interval coverage with narrower average intervals.

Estimated logistic PINN parameters
Model	r	K	σ
Deterministic logistic PINN	0.035062	52.3350	—
Stochastic logistic PINN/SDE	0.036806	52.2809	0.017245

The stochastic PINN adds a diffusion estimate, σ, while keeping similar logistic growth parameters.

1.Baltazar-Larios, F., Delgado-Vences, F. & Ornelas, A. Parameter estimation and model selection for stochastic differential equations for biological growth. Environ. Ecol. Stat. 32, 195–227 (2025).

2.Baltazar-Larios, F., Delgado-Vences, F., Diaz-Infante, S. & Gomez, E. L. Statistical inference for a stochastic generalized logistic differential equation. Commun. Nonlinear Sci. Numer. Simul. 108261 (2024) doi:10.1016/j.cnsns.2024.108261.

3.Baltazar-Larios, F., Delgado-Vences, F. & Diaz-Infante, S. Maximum likelihood estimation for a stochastic SEIR system with a COVID-19 application. Int J Comput Math 1–23 (2022) doi:10.1080/00207160.2022.2148316.

Gracias!

https://slides.com/sauldiazinfantevelasco/seminario-datascience-uam-2026-06-18

https://github.com/SaulDiazInfante/kefir-models.git

s aul.diazinfante@unison.mx

Seminario de Ciencia de Datos, UAM Iztapalapa Junio 18, 2026

Aprendizaje automático para dinámicas estocásticas en bioprocesos:

Contents

01

02

03

04

05

Modeling transition:

ODE → SDE

Numerical approximations for SDEs

Taylor explicit methods

Discrete Brownian motion

References

Numerical Solvers for Stochastic Differential Equations

References

References

Scientific Machine Learning (SciML)

State-of-the-Art SDE Software for SciML

Solving With PINNs

The Challenge: Simulating SDEs

Solving a SDE: classic numerics vs PINNs

Benchmark Example

References

Neural ODEs and Neural SDEs

Neural ODEs

Adjoint Method for Multi-Label Loss Functions

Nerural-SDE

Backward reconstruction of the state

References: Physics-Informed Neural Networks & UQ

References: Neural ODEs & SDEs

Example with real data

Kefir Water production

NODE fit

Evaluation Criteria

Water Kefir Model Comparison

Stochastic Dynamics and PINN Parameters