基于贝叶斯神经网络的Bratu方程
Inference of Bratu’s Equation Based on Bayesian Neural Networks

Bratu方程在描述热传导过程中的温度分布方面扮演着关键角色。本研究通过在一维区间内部署分段传感器阵列，采集了相应的温度数据。在充分考虑各传感器测量误差的基础上，我们利用贝叶斯推断方法对导体的整体温度分布进行了分析。研究首先基于采集到的数据，在不同水平的噪声干扰下进行推断，旨在通过贝叶斯神经网络获得与测试数据点相匹配的温度分布。随后，我们引入了描述热传导特性的Bratu方程，以期获得与该方程相符的温度分布，并据此通过贝叶斯神经网络进行进一步的推断。研究结果表明，相较于仅依赖于数据本身的推断，结合Bratu方程的贝叶斯推断能够显著提高预测的准确性和可靠性。这一发现强调了在热传导问题中，将物理模型与数据驱动方法相结合的重要性。
The Bratu equation plays a pivotal role in describing the temperature distribution during heat conduction. In this study, we deployed a segmented sensor array within a one-dimensional interval to collect corresponding temperature values. Considering the measurement errors of each sensor, we utilized Bayesian inference to analyze the overall temperature distribution of the conductor. Initially, we performed inference based on the collected data under various levels of noise interference, aiming to obtain a temperature distribution that aligns with the test data points through a Bayesian neural network. Subsequently, we incorporated the Bratu equation, which describes the characteristics of heat conduction, in the hope of achieving a temperature distribution that conforms to this equation and further inferred using the Bayesian neural network. The results indicated that the Bayesian inference combined with the Bratu equation significantly outperformed inference based solely on data. This finding underscores the importance of integrating physical models with data-driven approaches in heat conduction problems.