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Analytic regression of Feynman integrals from high-precision numerical sampling
This paper presents a novel bottom-up approach to analytically regress Feynman integrals from high-precision numerical sampling, leveraging lattice reduction techniques. It successfully extracts exact rational coefficients for complex integrals (up to 3-loops) by combining numerical evaluation tools (AMFLOW) with analytical knowledge of function spaces (SOFIA). The method demonstrates a trade-off between sampling points and required precision, offering a robust alternative to traditional matrix inversion or PSLQ for deriving exact analytic forms in theoretical physics.
Executive Impact
Leveraging AI for precise scientific and engineering challenges yields significant breakthroughs, directly translating to competitive advantages and accelerated innovation.
100+
Precision Achieved (digits)
3
Max Loops Analyzed
1300+
Basis Functions Handled
Deep Analysis & Enterprise Applications
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The core methodology combines high-precision numerical evaluation of Feynman integrals with analytical knowledge of the expected function space. Lattice reduction is then employed to deduce the exact rational coefficients.
Enterprise Process Flow
High-Precision Numerical Integration (AMFLOW)
→
Singularity Analysis & Basis Generation (SOFIA)
→
Numerical Sampling (f(x), Bi(x) data)
→
Lattice Reduction Algorithm
→
Exact Analytic Coefficient Recovery
| Method | Pros | Cons |
|---|---|---|
| Matrix Inversion |
|
|
| PSLQ |
|
|
| Lattice Reduction (LLL) |
|
|
An in-depth analysis of the scaling behavior of lattice reduction, exploring the trade-offs between the number of data points, basis functions, and required numerical precision. Optimizing point selection can further reduce precision needs.
dmin ~ n/p + d0
Precision-Sampling Trade-off Formula
Optimizing Point Selection for Efficiency
The paper demonstrates that strategically choosing sampling points, rather than random selection, can significantly reduce the minimum number of digits of precision required for a successful fit. For an example with 15 GPLs and 15 points, gradient descent optimization reduced the required precision by 1-2 digits, from ~10 digits to ~7 digits. This is crucial for applications with high computational costs for increasing precision.
The techniques are applicable beyond Feynman integrals to any problem requiring exact analytic reconstruction from high-precision numerical data where the function space is understood.
| Feature | Benefit |
|---|---|
| Exact Analytic Results |
|
| Robustness |
|
| Trade-off Control |
|
Beyond GPLs
Extensible to Elliptic Polylogarithms & Other Functions
Quantify Your AI Impact
Estimate the potential annual cost savings and hours reclaimed by implementing advanced analytic regression and AI-driven insights in your enterprise operations.
Estimated Annual Savings
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Hours Reclaimed Annually
0
Your AI Implementation Roadmap
A structured approach to integrate high-precision analytic regression into your enterprise workflows.
Phase 1: Discovery & Data Preparation
Identify target analytic problems, gather high-precision numerical data, and define potential function spaces.
Phase 2: Basis Function Generation
Utilize tools like SOFIA to construct the relevant alphabet and basis functions (e.g., GPLs) for your specific problem.
Phase 3: High-Precision Evaluation & Sampling
Employ numerical integration tools (e.g., AMFLOW) to generate sufficiently precise data samples for the target function and basis functions.
Phase 4: Analytic Regression & Validation
Apply lattice reduction (e.g., LLL) to recover exact rational coefficients. Validate the derived analytic expressions against additional data or theoretical constraints.
Phase 5: Integration & Operationalization
Integrate the derived analytic solutions into existing enterprise systems and processes for enhanced accuracy and new insights.
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