Learning Feynman integrals from differential equations with neural networks

Abstract

We perform an exploratory study of a new approach for evaluating Feynman integrals numerically. We apply the recently-proposed framework of physics-informed deep learning to train neural networks to approximate the solution to the differential equations satisfied by the Feynman integrals. This approach relies neither on a canonical form of the differential equations, which is often a bottleneck for the analytical techniques, nor on the availability of a large dataset, and after training yields essentially instantaneous evaluation times. We provide a proof-of-concept implementation within the PyTorch framework, and apply it to a number of one- and two-loop examples, achieving a mean magnitude of relative difference of around 1% at two loops in the physical phase space with network training times on the order of an hour on a laptop GPU.

33 pages, 12 figures, 3 tables, 2 appendices; published version

References in corpus (81)

• Ensemble deep learning: A review
• The total top quark pair production cross-section at hadron colliders through O(alpha_S^4)
• Multiloop integrals in dimensional regularization made simple
• Asymptotic expansion of Feynman integrals near threshold
• Differential Equations for Two-Loop Four-Point Functions
• A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks
• An automatized algorithm to compute infrared divergent multi-loop integrals
• Connection between Feynman integrals having different values of the space-time dimension
• The Path to Proton Structure at One-Percent Accuracy
• Automatized analytic continuation of Mellin-Barnes integrals
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Analytical Result for Dimensionally Regularized Massless On-shell Double Box
• Non-planar massless two-loop Feynman diagrams with four on-shell legs
• Efficient training of physics-informed neural networks via importance sampling
• Lectures on differential equations for Feynman integrals
• pySecDec: a toolbox for the numerical evaluation of multi-scale integrals
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• FIESTA 4: optimized Feynman integral calculations with GPU support
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop
• Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)
• A Systematic and Efficient Method to Compute Multi-loop Master Integrals
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• Resolution of singularities for multi-loop integrals
• The massless higher-loop two-point function
• On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
• AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals
• Problems of the Strategy of Regions
• FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions
• FIESTA 2: parallelizeable multiloop numerical calculations
• A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec
• Foundation and generalization of the expansion by regions
• Numerical evaluation of multi-loop integrals for arbitrary kinematics with SecDec 2.0
• From loops to trees by-passing Feynman's theorem
• Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections
• Machine Learning and LHC Event Generation
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• FIESTA5: numerical high-performance Feynman integral evaluation
• Multiloop corrections for collider processes using auxiliary mass flow
• A non-planar two-loop three-point function beyond multiple polylogarithms
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• Loop Tree Duality for multi-loop numerical integration
• Analytic Continuation of Massless Two-Loop Four-Point Functions
• Elliptic polylogarithms and Feynman parameter integrals
• Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals
• Solving differential equations for Feynman integrals by expansions near singular points
• HandyG -- rapid numerical evaluation of generalised polylogarithms in Fortran
• How to choose master integrals
• A factorisation-aware Matrix element emulator
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• A geometric method of sector decomposition
• Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case
• Analytic representation of all planar two-loop five-point Master Integrals with one off-shell leg
• Optimising simulations for diphoton production at hadron colliders using amplitude neural networks
• Tropical Monte Carlo quadrature for Feynman integrals
• A guide for deploying Deep Learning in LHC searches: How to achieve optimality and account for uncertainty
• Targeting Multi-Loop Integrals with Neural Networks
• Massive vector form factors to three loops
• Tropical Feynman integration in the Minkowski regime
• Numerical Scattering Amplitudes with pySecDec
• Using neural networks for efficient evaluation of high multiplicity scattering amplitudes
• Scattering Amplitudes in Quantum Field Theory
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions
• A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $\overline{\mathrm{MS}}$-pole quark mass relation
• Learning Specialized Activation Functions for Physics-informed Neural Networks
• Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel
• Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED
• A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop
• One-loop matrix element emulation with factorisation awareness
• Unweighting multijet event generation using factorisation-aware neural networks
• MB Tools reloaded
• Two-loop helicity amplitudes for $H+$jet production to higher orders in the dimensional regulator
• Evaluation of multi-loop multi-scale Feynman integrals for precision physics
• Multi-variable Integration with a Neural Network
• $\epsilon$-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process $e^{+}{e}^{-}\to 2(Q\bar{Q})$
• Feynman Integrals from Positivity Constraints
• Multi-variable integration with a variational quantum circuit