(675g) Drw-Bo: A Bayesian Framework for Parameter Estimation for Fractional Richards Equation with Applications in Precision Agriculture

Precision modeling and forecasting of soil moisture are essential for implementing smart irrigation systems and mitigating agricultural drought. Agro-hydrological models, which describe irrigation, precipitation, evapotranspiration, runoff, and drainage dynamics in soil, are widely used to simulate the root-zone (top 1m of soil) soil moisture content. Most existing agro-hydrological models incorporate the standard Richards equation [1], a highly nonlinear, degenerate elliptic-parabolic partial differential equation (PDE) with first order time derivative. However, case studies have shown that standard Richards equation alone may not be able to capture the water flow dynamics in porous media of fractal structure [2]. Additionally, recent experimental studies have demonstrated that water flow dynamics in soil exhibit anomalous non-Boltzmann scaling behavior [3]. To address these practical complications, fractional Richards equation, which employs a fractional order time derivative, is suggested as a generalization to the standard Richards equation to more model the complex water flow dynamics in soil. Such generalization brings computational challenge when it comes to solving the fractional Richards equation numerically. Recently, we have developed a data-facilitated numerical framework to solve fractional Richards equation accurately [4]. This state-of-the-art solver, which is referred to as the DRW solver, extends our recent progress in solving the standard Richards equation with superior numerical stability and accuracy [5].

In this talk, we build upon our DRW framework and explore techniques to solve the inverse problem of fractional Richards equation, i.e., estimating the soil properties and fractional order given actual root-zone soil moisture measurements. Conventional gradient-based optimization techniques face challenges in our problem due to the fractional nature of the PDE model. Instead, we focus on gradient free methods [6] to solve the inverse problem. In particular, we develop a novel Bayesian optimization (BO) framework, namely DRW-BO, to systematically perform parameter estimation and uncertainty quantification in the DRW framework. We will show the synergistic strength of our DRW-BO framework in solving the inverse problem by comparing our results with those using traditional optimization techniques, including genetic algorithm [7], and particle swarm optimization [8], as well as with other BO frameworks, such as the BO-PINN algorithm [9] using actual root-zone soil moisture data collected experimentally.

References

[1] L.A. Richards, Capillary conduction of liquids through porous mediums, Physics, 1931, 1(5): 318-333.

[2] Y. Pachepsky, D. Timlin, Water transport in soils as in fractal media, Journal of Hydrology, 1998, 204: 98-107.

[3] R. Liu, W.M. Ye, Y.J. Cui, H.H. Zhu, Q. Wang, Water infiltration and swelling pressure development in GMZ bentonite pellet mixtures with consideration of temperature effects, Engineering Geology, 2022, 305: 106718.

[4] Z. Song, Z. Jiang, A Computationally Efficient Data-Driven Framework for Solving Water Flow Dynamics in Soil Via Fractional Diffusion Model, AIChE Annual Meeting, 2023.

[5] Z. Song, Z. Jiang, A Data-facilitated Numerical Method for Richards Equation to Model Water Flow Dynamics in Soil, arXiv preprint arXiv:2310.02806, 2023.

[6] J. Mockus, The Bayesian approach to global optimization. System Modeling and Optimization: Proceedings of the 10th IFIP Conference, New York City, USA, August 31–September 4, 1981. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005.

[7] S. Zhou, J. Cao, Y. Chen, Genetic algorithm-based identification of fractional-order systems, Entropy, 2013, 15(5): 1624-1642.

[8] M.I. Romashchenko, V.O. Bohaienko, T.V. Matiash, V.P. Kovalchuk, A.V. Krucheniuk, Numerical simulation of irrigation scheduling using fractional Richards equation, Irrigation Science, 2021, 39(3):385-96.

[9] M. Rautela, S. Gopalakrishnan, J. Senthilnath, Bayesian optimized physics-informed neural network for estimating wave propagation velocities, arXiv preprint arXiv:2312.14064, 2023.