A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modelling). When complex dynamical systems are considered, such as partial differential equations, this task may become challenging or ill-posed. In this work, a linear parabolic equation is considered as a model for protein transcription from MRNA. The objective is to estimate jointly the differential operator coefficients, namely the rates of diffusion and self-regulation, as well as a functional source. The recent Bayesian methodology for infinite dimensional inverse problems is applied, providing a unique posterior distribution on the parameter space continuous in the data. This posterior is then summarized using a Maximum a Posteriori estimator. Finally, the theoretical solution is illustrated using a state-of-the-art MCMC algorithm adapted to this non-Gaussian setting.

实验科学中的一个常见任务是将数学模型拟合到现实世界的测量中，以提高对自然现象的理解（逆向工程或逆向建模）。当考虑复杂的动力系统时，例如偏微分方程，这项任务可能变得具有挑战性或不适定。在这项工作中，线性抛物线方程被认为是从 mRNA 转录蛋白质的模型。目标是联合估计微分算子系数，即扩散和自我调节的速率，以及函数源。应用了最近用于无限维逆问题的贝叶斯方法，在数据中连续的参数空间上提供了独特的后验分布。然后使用最大后验估计量总结这个后验。最后，