Solving differential equations for chemical kinetics using quantics tensor train

In a chemical system, multiple elementary reactions, each with significantly different reaction rates, often occur. In such systems, when solving differential equations numerically, the time step of numerical simulations is confined to a very small time scale, resulting in a substantial increase in the computational cost. This issue becomes particularly pronounced when nonlinear terms are present, as it becomes difficult to handle long and short time scales separately. Given these points, developing a method that can efficiently handle different time scales simultaneously is highly desirable.

Quantics tensor train (QTT), a type of tensor network, can adaptively compress a scalar-valued function according to the amount of correlation between different length scales [1, 2]. Recently, QTT has been applied to various fields, such as quantum field theory [3], and directly solving differential equations, including those in fluid dynamics [4], chemical master equations [5], and so on.

This study considers compression in QTT format of nonlinear ordinary differential equations for chemical kinetics models. First, we verify a low-rank structure for the kinetics model of Escherichia coli core metabolism [6]  by simultaneously compressing the solution both in time and chemical species. Secondly, we propose an iterative method for solving the differential equations with good initial solutions of the time derivative of concentrations from short to long time scales. 

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[3] H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, A. Kauch, Phys. Rev. X 13, 021015 (2023)

[4] N.Gourianov, M. Lubasch, S. Dolgov, Q. Y. van den Berg, H. Babaee, P. Givi, M. Kiffner, D. Jaksch, Nat. Comput. Sci. 2, 30 (2022)

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[6] A. Khobayari, A. R. Zomorrodi, J.C. Liao, C. D. Maranas, Metabolic Engineering 25, 50 (2014)