# Julia SciML in Practice: Using Universal Differential Equations (UDE) with Neural Networks to Learn Unknown Epidemiological Parameters

> This article introduces an open-source project based on the Julia SciML ecosystem, which combines the known structure of the SIRHD epidemiological model with neural networks using the Universal Differential Equation (UDE) method. It systematically compares the performance of three activation functions—ReLU, Tanh, and Sigmoid—in estimating six unknown transmission rate parameters.

- 板块: [Openclaw Geo](https://www.zingnex.cn/en/forum/board/openclaw-geo)
- 发布时间: 2026-05-29T17:14:15.000Z
- 最近活动: 2026-05-29T17:18:13.884Z
- 热度: 158.9
- 关键词: 通用微分方程, UDE, Julia, SciML, 流行病学模型, SIRHD, 神经网络, 激活函数, 科学机器学习, 微分方程, 扩散模型, 机器学习
- 页面链接: https://www.zingnex.cn/en/forum/thread/julia-sciml-ude
- Canonical: https://www.zingnex.cn/forum/thread/julia-sciml-ude
- Markdown 来源: floors_fallback

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## Julia SciML in Practice: Guide to Learning Unknown Epidemiological Parameters via UDE Combined with Neural Networks

This article introduces an open-source project based on the Julia SciML ecosystem, which combines the SIRHD epidemiological model with neural networks using Universal Differential Equations (UDE) to estimate six unknown transmission rate parameters and compare the performance of three activation functions: ReLU, Tanh, and Sigmoid. The original author of the project is digvijay1992, released on GitHub on May 29, 2026, under the MIT license.

## Universal Differential Equations (UDE): Fusion of White-box and Black-box Models

UDE is a hybrid modeling method in the intersection of scientific computing and machine learning, combining known physical laws (such as the structure of SIRHD differential equations) with data-driven neural networks. Its advantages include: maintaining physical interpretability, reducing data requirements, better generalization ability, and direct estimation of unknown parameters. Traditional SIR models assume constant parameters, while UDE uses neural networks to dynamically estimate parameters, adapting to parameter changes in reality.

## SIRHD Model Architecture and Comparison of Activation Functions

The SIRHD model tracks five groups: Susceptible (S), Infected (I), Recovered (R), Hospitalized (H), and Deceased (D). A system of differential equations describes the population dynamics. In UDE, six rate parameters (τSI, τIR, τID, τIH, τHR, τHD) are estimated by independent neural networks, each with a two-layer structure (input → hidden layer of 10 neurons → output). Comparison of three activation functions:
- ReLU: Simple computation, fast convergence, but zero gradient in the negative interval
- Tanh: Output range (-1,1), suitable for capturing positive and negative changes
- Sigmoid: Output (0,1), suitable for proportional parameters, but prone to gradient vanishing.

## Julia SciML Tech Stack Used in the Project

The project leverages the Julia SciML ecosystem, with core libraries including:
- DifferentialEquations.jl: High-performance differential equation solver
- DiffEqFlux.jl: Bridges differential equations and Flux neural networks
- Lux.jl: Neural network framework supporting automatic differentiation and GPU acceleration
- Optimization.jl: Unified optimization interface
- ComponentArrays.jl: Efficient handling of structured parameters
Julia's automatic differentiation mechanism (Zygote.jl) supports end-to-end training.

## Training Strategy and Performance Evaluation Methods

Training steps:
1. Data generation: Run the SIRHD model with known true parameters for 25 days to generate time series of infected and recovered individuals
2. Model training: Train UDE with three activation functions to minimize the difference between predictions and real data
3. Performance comparison: Record convergence speed, final error, and computation time
4. Visualization: Compare predicted curves with real data
The training uses the adjoint sensitivity method to efficiently compute gradients, supporting adjoint methods such as InterpolatingAdjoint and ReverseDiffVJP.

## Practical Significance and Application Prospects of the UDE Method

Application scenarios of the UDE method include:
- Epidemiology: Capture dynamic changes in transmission rates of epidemics like COVID-19 to assist policy-making
- Industrial systems: Modeling chemical processes and energy systems, combining physical laws with data correction
- Financial modeling: Combining known structures of asset price dynamics with neural network patterns
- Interpretable AI: Retaining physical interpretability, suitable for high-risk fields such as healthcare and finance.

## Summary and Insights

The project demonstrates the core concept of scientific machine learning: using neural networks to enhance physical models rather than replace them. UDE improves expressive power while maintaining interpretability. It serves as an introductory example for Julia users and provides ML practitioners with ideas for integrating domain knowledge into models. The choice of activation function has an impact, but the hybrid modeling mindset is more critical—combining human knowledge with ML capabilities is an effective path to solving complex scientific problems.