FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs

TL;DR

FiniteFlow presents a framework to reconstruct analytic multivariate rational functions from numerical evaluations over finite fields using dataflow graphs. It combines a library of finite-field numerical algorithms with graph-based composition and multivariate reconstruction, enabling flexible, parallelizable workflows for complex algebraic calculations. The approach is demonstrated across key high-energy-physics tasks, including IBP reduction to master integrals, differential equations for master integrals, integrand reduction, and symbol derivation, complemented by a proof-of-concept C++ implementation with a Mathematica interface. By shifting heavy symbolic manipulation into modular, parallel finite-field computations and subsequent reconstruction, FiniteFlow reduces intermediate expression growth and broadens the accessible toolbox for precision multi-loop computations.

Abstract

Complex algebraic calculations can be performed by reconstructing analytic results from numerical evaluations over finite fields. We describe FiniteFlow, a framework for defining and executing numerical algorithms over finite fields and reconstructing multivariate rational functions. The framework employs computational graphs, known as dataflow graphs, to combine basic building blocks into complex algorithms. This allows to easily implement a wide range of methods over finite fields in high-level languages and computer algebra systems, without being concerned with the low-level details of the numerical implementation. This approach sidesteps the appearance of large intermediate expressions and can be massively parallelized. We present applications to the calculation of multi-loop scattering amplitudes, including the reduction via integration-by-parts identities to master integrals or special functions, the computation of differential equations for Feynman integrals, multi-loop integrand reduction, the decomposition of amplitudes into form factors, and the derivation of integrable symbols from a known alphabet. We also release a proof-of-concept C++ implementation of this framework, with a high-level interface in Mathematica.

Figures (6)

• Figure 1: A node in a dataflow graph, where arrows represent lists of values and nodes represent numerical algorithms. In our implementation, a node can take zero or more incoming arrows as input and has exactly one outgoing arrow as output.
• Figure 2: A node representing the evaluation of a list of rational functions. It takes one list $\mathbf{z}$ as input, interpreted as the list of arguments of the functions, and returns the values of the functions evaluated at $\mathbf{z}$.
• Figure 3: A node representing a linear solver. It takes as input a list of parameters $\mathbf{z}$ and returns the coefficients of the solution defined in Eq. \ref{['eq:linsolsol']} and \ref{['eq:linsolcmat']}.
• Figure 5: Two dataflow graphs representing the reduction of a scattering amplitude to master integrals. The graph on the right has two additional nodes chaining to the coefficients of the IBP solutions an identity matrix, which represents the (trivial) reduction of the master integrals themselves. These nodes are needed when the masters can also appear on the r.h.s. of the unreduced amplitude in Eq. \ref{['eq:ampfunred']}.
• Figure 6: Two graphs which, combined, compute the $\epsilon$ expansion of the coefficients of scattering amplitudes in terms of special functions. In the first graph $G_1$, $A_{j}$ represents the calculation of the coefficients of the master integrals presented in section \ref{['sec:reduct-mast-integr']} and fig. \ref{['fig:amp']}. The graph $G_2$ then takes the graph $G_1$ as subgraph in one of its nodes, which computes its Laurent expansion in $\epsilon$.