Interpolating amplitudes

Víctor Bresó; Gudrun Heinrich; Vitaly Magerya; Anton Olsson

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Interpolating amplitudes

Víctor Bresó, Gudrun Heinrich, Vitaly Magerya, Anton Olsson

arXiv:2412.09534v2Dec 12, 2024hep-ph

TL;DR

Numerical multi-loop amplitude evaluations can be unfeasible at individual phase-space points because realistic Monte Carlo calculations require millions of points. The paper benchmarks polynomial, B-spline, spatially adaptive sparse-grid, multilayer-perceptron, and Lorentz-Equivariant Geometric Algebra Transformer interpolation methods on amplitude-related test functions. For five-dimensional test functions, machine learning significantly outperforms classical methods at 10^-2 to 10^-4 approximation error, while sparse grids are best for very high precision in the lower-dimensional case.

Problem
Numerical multi-loop amplitude evaluations can be unfeasible at individual phase-space points because realistic Monte Carlo calculations require millions of points.
Method
The paper benchmarks polynomial, B-spline, spatially adaptive sparse-grid, multilayer-perceptron, and Lorentz-Equivariant Geometric Algebra Transformer interpolation methods on amplitude-related test functions.
Results
For five-dimensional test functions, machine learning significantly outperforms classical methods at 10^-2 to 10^-4 approximation error, while sparse grids are best for very high precision in the lower-dimensional case.
Takeaways & Limitations
Machine learning is the most promising practical tool for high-dimensional amplitude interpolation, supporting feasible interpolation frameworks for current and upcoming collider studies.
Takeaways & Limitations
The selected L1 error prioritizes total-cross-section precision, so it does not guarantee equal relative precision in low-contribution phase-space tails.

Abstract

from arXiv · show

The calculation of scattering amplitudes at higher orders in perturbation theory has reached a high degree of maturity. However, their usage to produce physical predictions within Monte Carlo programs is often precluded by the slow evaluation of two- and higher-loop virtual amplitudes, particularly those calculated numerically. As a remedy, interpolation frameworks have been successfully used for amplitudes depending on up to two kinematic invariants. For amplitude interpolation with more variables, such as the five dimensions of a 2 -> 3 phase space, efficient and reliable solutions are sparse. This work aims to pave the way for using amplitude interpolation in higher-dimensional phase spaces by reviewing state-of-the-art interpolation methods, and assessing their performance on a selection of 2 -> 3 scattering amplitudes. Specifically, we investigate interpolation methods based on polynomials, splines, spatially adaptive sparse grids, and neural networks (multilayer perceptron and Lorentz-Equivariant Geometric Algebra Transformer), all under the constraint of limited obtainable data. Our additional aim is to motivate further studies of the interpolation of scattering amplitudes among both physicists and mathematicians.

1 Introduction

Higher-order scattering-amplitude calculations are increasingly precise, but numerically evaluated multi-loop amplitudes remain difficult to use in Monte Carlo predictions. This work evaluates multidimensional interpolation methods as a practical route toward five-dimensional 2 → 3 phase spaces.

1 Introduction: Numerical multi-loop amplitudes become increasingly difficult as the number of external legs or particle masses grows, limiting analytic approaches.The underlying function classes and algebraic complexity become rapidly more complicated.
1 Introduction: Interpolation can precompute expensive amplitude values and approximate them at phase-space points required by Monte Carlo programs.The paper presents this as a first step toward using numerically evaluated, non-analytic multi-loop amplitudes in phenomenology.
1 Introduction: 2 → 3 interpolation requires five-dimensional phase-space treatment, unlike existing grid applications for 2 → 2 amplitudes with two kinematic variables.This higher dimensionality makes achieving precision substantially more challenging.
1 Introduction: The paper reviews state-of-the-art multidimensional interpolation methods and assesses their practical performance for scattering-amplitude applications.It connects mathematical interpolation literature with physics use cases that have mostly remained below five dimensions.
1 Introduction: The study covers polynomial interpolation, B-splines, sparse grids, and machine-learning techniques, with results organized around approximation error and training data.The methods are presented across Sections 3–6 and benchmarked through error plots.

2 Setting the stage

The paper frames interpolation as approximating an expensive amplitude component over phase space while controlling an error measure relevant to cross sections. It emphasizes that the chosen error definition determines which regions receive precision and how methods are evaluated.

2.2 Our goal: 2 → 2, 2 → 3, and 2 → 4 processes require 2, 5, and 9 phase-space parameters, respectively.The parameters are chosen so the physical region becomes a hypercube.
2.2 Our goal: The goal is to approximate a slow or expensive amplitude component from as few sampled phase-space points as possible.The approximation should be accurate enough while minimizing costly evaluations.
2.3 How to define the approximation error: The selected approximation error compares differential cross sections from the exact and interpolated amplitudes using an L1 distance.This treats the differential cross section as a probability density.
2.3 How to define the approximation error: The L1 error is variable-independent and weights phase-space regions according to their contribution to the total cross section.The weighting factor w determines the relative importance assigned to different regions.
2.3 How to define the approximation error: The chosen error measure prioritizes total-cross-section precision over uniform relative precision in sparsely populated high-energy tails.An alternative maximum relative error would treat all differential-distribution bins equally but would focus effort on regions with few expected data points.
2.4 How to prioritize different phase-space regions: Importance information can be incorporated through weighted targets, clustered samples, adaptive refinement, or variable transformations.The paper studies these options across its interpolation methods, but reports limited gains from the transformations in its testing.
2.5 Test functions: The benchmarks use five test functions related to t̄tH and Higgs+jet production, including a five-dimensional t̄tH phase space.The phase-space density is modeled with a generic form approximating parton-distribution-function behavior.
2.7 How to evaluate the approximation error: Low-discrepancy Sobol sampling gives only marginal error-convergence improvements over random Monte Carlo sampling, with errors stabilizing after about m ∼1000 testing samples.The comparison is shown for sparse-grid interpolation of f2.

3 Polynomial interpolation

Polynomial interpolation constructs a polynomial through sampled nodes, with node placement controlling approximation error. Chebyshev-based choices avoid the boundary instability associated with naive equidistant nodes, while multidimensional use faces the curse of dimensionality.

3 Polynomial interpolation: Polynomial interpolation constructs a degree n − 1 polynomial that passes through n interpolation nodes.The paper presents both Lagrange and barycentric forms, choosing the latter for stable and simple evaluation.
3 Polynomial interpolation: Equidistant nodes can produce the Runge phenomenon, worsening approximation error near domain boundaries as polynomial degree increases.Node placement must therefore be selected to control the interpolation error over the domain.
3 Polynomial interpolation: Nonlinear rational interpolation such as AAA is effective in one dimension, but its multidimensional generalizations are not developed enough for this study.Consequently, the paper focuses on polynomial interpolation for this method category.

Chebyshev polynomials

Chebyshev polynomials connect interpolation nodes to an orthogonal polynomial basis and a Fourier representation. Chebyshev–Lobatto nodes add nesting useful for adaptive grids, but their endpoints can be impractical for scattering amplitudes.

Chebyshev polynomials: Chebyshev nodes of the first kind are zeros of the corresponding Chebyshev polynomials, which are orthogonal under their weight.The node-to-coefficient transformation can be viewed as a discrete cosine transform.
Chebyshev polynomials: The barycentric form is preferred for interpolation because monomial and Chebyshev-coefficient forms suffer rounding errors for n ≳40.This preference reflects numerical stability during evaluation.
Chebyshev polynomials: Chebyshev–Lobatto nodes include extrema and endpoints and are nested, so larger adaptive grids can reuse smaller-grid evaluations.The nesting comes at the cost of a nonuniform node-product bound.
Chebyshev polynomials: Endpoint inclusion can be impractical for scattering amplitudes because f2 cannot be evaluated exactly at x2 = 0 and evaluation near x1 = 1 is typically slower.This limits the direct usefulness of Chebyshev–Lobatto constructions in the cited amplitude setting.

Gauss nodes

The discussion contrasts Gauss-type nodes with Chebyshev nodes and then extends Chebyshev interpolation to multiple dimensions, where convergence slows with dimension.

Gauss nodes: Gauss nodes are defined by Legendre-polynomial zeros, while Gauss–Lobatto nodes are defined by Legendre-polynomial extrema.
Gauss nodes: Gauss quadrature is exact for polynomials through degree 2n −1, whereas Clenshaw–Curtis quadrature reaches degree n −1.Despite this difference, the practical approximation errors are reported to be very close.
Gauss nodes: Because Gauss nodes are harder to compute and offer nearly identical practical errors, the study does not consider them further.
Multiple dimensions: The multidimensional construction uses an outer tensor product of Chebyshev nodes, allowing different node counts n_i for each dimension.The interpolant can equivalently be formed by nested one-dimensional interpolation or a tensor-product Chebyshev decomposition.
Multiple dimensions: Tensor-product interpolation converges progressively more slowly as dimension d increases, reflecting the curse of dimensionality.The relevant per-dimension node counts n_i scale with the available data budget in a dimension-dependent way.

3.3 Dimensionally adaptive grid

The dimensionally adaptive-grid analysis uses coefficient structure to tune sampling across dimensions, while motivating non-tensor-product bases to reduce diagonal oversampling.

3.3 Dimensionally adaptive grid: The f1 coefficient slices reveal discrete symmetries in dimensions 3, 4, and 5, producing a checkerboard pattern with zero basis coefficients.
3.3 Dimensionally adaptive grid: Coefficient scaling suggests the sampling ratio n1 : n2 : n3 : n4 : n5 ≈2 : 4 : 2 : 3 : 1 to keep directional coefficients comparably resolved.
3.3 Dimensionally adaptive grid: Choosing the best dimension ratio separately for each data budget would improve results, but that optimization is only available a posteriori.
3.3 Dimensionally adaptive grid: Figure 4 compares approximation errors across sampling ratios, interpolated functions, symmetry handling, and weighting choices as data evaluations increase.The plotted evaluation count accounts for each function’s symmetries, and grey curves show later methods’ best results.
3.3 Dimensionally adaptive grid: The coefficient slices also show smaller values toward the upper-right diagonals, indicating that cardinal-direction tuning can still oversample diagonal regions.
3.4 Beyond the full grid: A non-tensor-product alternative truncates the multi-index set with an Lp condition and selects nodes greedily in Leja order.This removes upper-right multi-index regions and uses the first n points ordered by successive maximization of the product of distances.

3.5 Effects of noisy data

The noisy-data study tests how interpolation error changes when amplitude evaluations contain multiplicative noise, using deliberately high noise levels to make effects visible.

3.5 Effects of noisy data: The analysis applies multiplicative noise to amplitude evaluations to assess interpolation with low-precision training values.The precision target matters because evaluating amplitudes more accurately generally requires more time.
3.5 Effects of noisy data: The noise levels are intentionally chosen to be very high so that the resulting effects are more visible.

3.6 Discussion

Polynomial interpolation is presented as a dependable default, but its multidimensional scaling, boundary singularities, and inflexible node sets constrain its use for amplitudes.

3.6 Discussion: Polynomial interpolation is a safe default because it converges exponentially for analytic one-dimensional functions and can be evaluated accurately with the barycentric formula.
3.6 Discussion: The tensor-product grid causes the curse of dimensionality, while nearby singularities and non-analytic boundaries worsen convergence.
3.6 Discussion: Polynomial node sets are inflexible: adding a few data points or smoothly extending an interpolant beyond a subset of phase space is difficult.Nested constructions roughly double the data count rather than permitting small incremental updates.
3.6 Discussion: For f5, including the weight helps substantially, whereas it slightly hinders other functions, possibly because the weight is non-analytic at boundaries.
3.6 Discussion: Domain reduction significantly helps f3 and f4, has little effect on f1 and f2, and slightly hinders f5, for which data duplication performs better.
3.6 Discussion: Cancelling the 1/sin2 θH factor modestly but noticeably improves the f5 interpolant, although subtracting the peak might improve it further.

4 B-spline interpolation

B-spline interpolation uses locally supported, non-negative basis functions arranged on tensor-product grids to approximate multidimensional amplitudes. Quadratic and cubic splines substantially outperform linear splines, but tensor-product scaling and boundary evaluations limit practical efficiency.

4.1 B-spline basis functions: B-spline basis functions are non-negative, form a partition of unity, have local support, and can be evaluated efficiently, especially with uniform knot vectors.
4.3 Discussion: Tensor-product B-splines require (n+1)^d interpolation nodes and have uncertain point requirements, making the reported results optimistic in practice.
4.3 Discussion: Quadratic or cubic B-splines significantly outperform linear splines for all five test functions.
4.3 Discussion: Cubic splines perform slightly better for f1 and f2, whereas quadratic splines are best for f3, f4, and f5.
4.3 Discussion: For f1 and f3, directly interpolating the amplitude is better, while for f5 including the weight helps with little training data but becomes worse with more data.
4.3 Discussion: B-splines require boundary evaluations, which can be problematic near amplitude singularities and may force shifting the interpolation domain.

5 Sparse grids

Sparse grids reduce full-grid growth by omitting highly refined multidimensional subspaces, while hierarchical surpluses enable spatial adaptivity. In the benchmarks, balanced refinement is the more reliable strategy, higher-degree bases often improve accuracy, and adaptive sparse grids scale better at high data counts.

5.1 Sparse-grid construction: Sparse grids select hierarchical subspaces with bounded summed levels, avoiding basis functions refined in every direction simultaneously and reducing high-dimensional point growth.The hierarchical construction uses local-support basis functions and surpluses that measure level-wise interpolation corrections, enabling local adaptivity.
5.2 Boundary treatment: Boundary treatment is a central constraint: adding boundary points can greatly increase point counts, while modified basis functions extrapolate from the outermost grid points.The basic hat-function basis only handles target functions that vanish at boundaries; higher-degree extensions may remain merely C0 continuous.
5.3 Spatially adaptive sparse grids: Balanced refinement is more reliable than greedy refinement because greedy refinement can get stuck in local structures and is more sensitive to training-data noise.Balanced refinement weights hierarchical surpluses by basis-function support volume, suppressing local deviations by construction.
5.5 Discussion: Increasing the degree improves all test functions for piecewise-polynomial and fundamental B-spline bases, whereas extended not-a-knot B-splines improve mainly f5 and can converge poorly for f3 and f4.The cubic and quintic extended not-a-knot cases perform poorly for f3 and f4 despite earlier favorable applications reported in.
5.5 Discussion: Interpolating ratios to leading-order amplitudes improves the low-data regime but worsens scaling relative to direct amplitude interpolation.Higher-degree basis functions appear more robust against this worse scaling.
5.5 Discussion: Spatially adaptive sparse grids show modest low-data gains but significantly better scaling than non-adaptive full-grid methods, producing a performance boost at high data regimes.They also allow training points to be added iteratively and evaluated during construction, supporting stopping at a desired precision or data budget.

6 Machine Learning Techniques

The paper evaluates neural-network approaches for amplitude interpolation, comparing Lorentz-aware inputs and architectures under limited-data conditions. Across the tested functions, neural methods are strongest at low-to-medium precision and in five dimensions, but eventually stagnate as training data and precision demands increase.

6.3 Discussion: MLP(s) and L-GATr perform similarly across all test functions, while MLP(x) performs consistently worse.MLP(s) is slightly better for f5 and f2 at low precision, but slightly worse for f4 at high precision.
6.3 Discussion: Neural-network improvement slows as training data increases, with performance eventually approaching a lower bound associated here mainly with training time.Longer MLP training substantially improves performance in the large-data limit, but the limitation requires further study.
6.3 Discussion: Machine-learning methods surpass other interpolation methods in the low-to-medium precision regime for f1–f4, especially for the more complicated higher-order corrections f2 and f4.They are overtaken only in the high-precision limit, after their results stop improving.
6.3 Discussion: Neural networks underperform for f5, the only two-dimensional test function, and their results show overfitting in the 10^3–10^4 data-point range under the reported long-training setup.The lower-dimensional setting provides less advantage for neural methods, while the degraded intermediate-range performance is attributed to overfitting.
6.3 Discussion: Training on uniform or unweighted samples produces similar neural-network performance across training-set sizes, enabling naive sampling while targeting physically motivated evaluation distributions.This distributional insensitivity is observed for the MLP in the reported comparison.
6.3 Discussion: Dividing by the leading-order amplitude yields a small but stable improvement for f2 and f4 in neural networks, unlike the disappearing advantage observed for polynomials and sparse grids with more data.For higher-order corrections, neural approaches benefit from the ratio at all training sizes, whereas some other methods benefit mainly in the low-data regime.

7 Conclusions

The conclusions find multidimensional amplitude interpolation feasible, with machine learning particularly promising for five-dimensional cases. Classical methods remain competitive for very high precision in lower dimensions, while the study excludes useful two-loop 2 → 3 examples because generating their training data was beyond scope.

7 Conclusions: The study uses t̄tH amplitudes on a five-dimensional phase space and Higgs+jet amplitudes on a two-dimensional phase space to benchmark polynomials, B-splines, adaptive sparse grids, and neural networks.Approximation-error scaling with the number of data points is the main performance measure, using physically motivated error metrics weighted by cross-section contribution.
7 Conclusions: Machine-learning techniques significantly outperform classical interpolation for five-dimensional test functions at 10^-2 to 10^-4 approximation error, sometimes by several orders of magnitude.For lower-dimensional functions, most methods perform adequately, but sparse grids are best when very high precision is required.
7 Conclusions: Machine-learning performance stagnates beyond a precision threshold, whereas classical methods scale smoothly, yet machine learning remains the most practical choice in high dimensions.The threshold is described as sufficiently high to preserve this practical advantage while motivating further investigation.
7 Conclusions: Training on a/aLO instead of the raw amplitude a improves interpolation for higher-order corrections, with neural networks benefiting at all training sizes.For some other methods, the improvement is significant only in the low-data regime.
7 Conclusions: Useful two-loop 2 → 3 examples are absent because generating their training data would have exceeded the scope of the paper.Simple two-loop examples were omitted because their low-dimensional phase spaces would make most interpolation methods perform well.
7 Conclusions: Adaptive methods and neural networks can be extended by adding training data, while spatially adaptive sparse grids allow single-point additions and assessment of poorly approximated regions.Extendibility matters because the data required to reach a target error is difficult to predict.

A.1 Phase space parametrisation for pp →t ¯t H

The t̄tH test functions use a five-dimensional phase-space parametrization built from three angular variables and two energy variables. The parametrization includes invariant-mass and threshold variables for describing the partonic kinematics.

A.1 Phase space parametrisation for pp →t̄tH: The f1–f4 test functions describe a 2 → 3 phase space for t̄tH production using three angular variables and two energy variables.
A.1 Phase space parametrisation for pp →t̄tH: The parametrization uses the Higgs polar angle, the t̄t solid-angle element, the Källén function, and invariant masses such as ŝ and st̄t.The Källén function depends on the relevant variables and masses.
A.1 Phase space parametrisation for pp →t̄tH: The threshold variable β^2 is zero at the t̄tH production threshold and approaches one in the high-energy limit.The threshold is located at s0 = (2mt + mH)^2.