Iterative tensor network transformations for element-wise evaluation of elementary and filtering functions

Details

Tensor networks, particularly tensor trains (TTs), are powerful for compressing large-scale scientific data, but have historically been limited to linear operations in the compressed domain. This paper introduces ITNTs (Iterative Tensor Network Transformations), a general algorithmic framework that enables element-wise evaluation of nonlinear functions on data encoded as tensor trains — without decompressing back to full representation. The framework is demonstrated in two domains: (1) computing reaction rates in a 3D reactive flow field by evaluating highly nonlinear elementary functions, enabling high-fidelity chemistry computation on compressed fluid simulation data; and (2) finding extrema in combinatorial optimization problems, specifically solving Max-SAT instances over configuration spaces of up to 2^70 states. Both demonstrations show that ITNT can operate on exponentially large datasets while keeping computational cost tractable. This is a foundational methods contribution bridging tensor networks and general-purpose data science. Practical impact for AI practitioners is indirect — tensor networks are not mainstream ML tools — but the ability to perform optimization over 2^70-size spaces is potentially relevant for large discrete search problems.