Learning to Unscramble Feynman Loop Integrals with SAILIR

David Shih

hep-phApr 6, 2026

Learning to Unscramble Feynman Loop Integrals with SAILIR

Authors: David Shih

Organizations: NHETC, Dept. of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA

Abstract

Integration-by-parts (IBP) reduction of Feynman integrals to master integrals is a key computational bottleneck in precision calculations in high-energy physics. Traditional approaches based on the Laporta algorithm require solving large systems of equations, leading to memory consumption that grows rapidly with integral complexity. We present SAILIR (Self-supervised AI for Loop Integral Reduction), a new machine learning approach in which a transformer-based classifier guides the reduction of integrals one step at a time in a fully online fashion. The classifier is trained in an entirely self-supervised manner on synthetic data generated by a scramble/unscramble procedure: known reduction identities are applied in reverse to build expressions of increasing complexity, and the classifier learns to undo these steps. When combined with beam search and a highly parallelized, asynchronous, single-episode reduction strategy, SAILIR can reduce integrals of arbitrarily high weight with bounded memory. We benchmark SAILIR on the two-loop triangle-box topology, comparing against the state-of-the-art IBP reduction code Kira across 16 integrals of varying complexity. While SAILIR is slower in wall-clock time, its per-worker memory consumption remains approximately flat regardless of integral complexity, in contrast to Kira whose memory grows rapidly with complexity. For the most complex integrals considered here, SAILIR uses only 40% of the memory of Kira while achieving comparable reduction times. This demonstrates a fundamentally new paradigm for IBP reduction in which the memory bottleneck of Laporta-based approaches could be entirely overcome, potentially opening the door to precision calculations that are currently intractable.

Figures & tables

Panel of Figure 1 — Figure 1: Architecture of the action classifier. All inputs are embedded to a common dimension ( $d_{\text{emb}}=256$ ); embedding layers are described in Appendix C . Expression terms and substitution histories are each processed by independent Transformer encoders. The expression Transformer produces per-term embeddings (keys/values for cross-attention, dashed) and pooled [CLS] + [TGT] tokens. The global state vector $\mathbf{h}$ concatenates the [CLS] , [TGT] , substitution, and sector representations. Action scores are computed via cross-attention (actions as queries, expression terms as keys/values) followed by a scoring MLP that combines the attended action representations with $\mathbf{h}$ .

Figure 2: Training and validation cross-entropy loss (left) and top-1 accuracy (right) as a function of epoch. The dashed line marks epoch 22, where the best validation loss is achieved.

$r$	$s$	Integral	$r$	$s$	Integral
10	4	$I[2,1,2,1,2,2,-4]$	10	6	$I[1,1,3,2,2,1,-6]$
11	4	$I[2,2,2,1,1,3,-4]$	11	6	$I[1,4,2,1,2,1,-6]$
12	4	$I[2,3,1,3,1,2,-4]$	12	6	$I[3,2,3,2,1,1,-6]$
13	4	$I[2,3,3,3,1,1,-4]$	13	6	$I[3,2,1,3,2,2,-6]$
10	5	$I[1,1,2,2,1,3,-5]$	10	7	$I[2,3,1,1,2,1,-7]$
11	5	$I[1,1,2,3,2,2,-5]$	11	7	$I[2,1,1,2,3,2,-7]$

Table 1: The 16 benchmark integrals, all in the top sector

(111111)_{2}

. For each

(r,s)

, the propagator indices were chosen randomly.

Figure 3: Kira vs. Sailir on the two-loop triangle-box topology, for

r\in\{10,\ldots,13\}

and

s\in\{4,\ldots,7\}

. Dashed lines with circles: Kira . Solid lines with squares: Sailir (ideal parallel time, per-worker memory). Colors indicate different values of

r

. Left : Time vs.

s

. Kira time grows rapidly with

s

; Sailir time grows more slowly, and the curves converge at high

s

. Right : Peak memory vs.

s

. Kira memory grows by nearly two orders of magnitude; Sailir per-worker memory remains approximately flat at

{\sim}

3 GB. The memory crossover occurs between

s=6

and

s=7

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Step	Target	$w_{\text{target}}$	Action	Seed	$w_{\max}$	Non-masters
						all	sector
1	$I[1,0,-\!1,1,2,0,0]$	$(4,1)$	$\partial_{k_{2}}\!\cdot\!p_{2}$	$I[1,0,-\!1,1,2,0,0]$	$\mathbf{(5,2)}$	9	8
2	$I[1,0,-\!1,1,3,-\!1,0]$	$(5,2)$	$\partial_{k_{2}}\!\cdot\!k_{1}$	$I[1,0,0,1,1,0,0]$	$(4,1)$	4	3
3	$I[1,-\!1,0,1,2,0,0]$	$(4,1)$	$\partial_{k_{2}}\!\cdot\!k_{2}$	$I[1,0,0,1,1,0,0]$	$(4,1)$	4	3
4	$I[1,0,0,1,2,0,-\!1]$	$(4,1)$	$\partial_{k_{1}}\!\cdot\!k_{1}$	$I[1,0,0,1,2,0,-\!1]$	$\mathbf{(5,1)}$	5	3
5	$I[1,0,0,2,2,0,-\!1]$	$(5,1)$	$\partial_{k_{1}}\!\cdot\!p_{1}$	$I[1,0,0,1,2,0,-\!1]$	$(5,1)$	6	3

Appendix

Table 2: Weight trajectory of a non-monotonic 8-step Sailir episode reducing

I[1,0,-\!1,1,2,0,0]

in sector

(1,0,0,1,1,0)

, starting at

w=(4,1)

. Each step applies the IBP/LI identity specified by the Action (operator) evaluated at the Seed, then solves for the Target.

w_{\text{target}}

: weight of the integral being eliminated.

w_{\max}

: maximum weight among non-master integrals in the target sector after the step. The last two columns count non-master integrals remaining in the expression: “all” across all sectors, and “sector” restricted to the target sector. Bold

w_{\max}

entries mark weight increases.

Step	$w_{\max}$	Valid	Non-masters
		actions	all	sector
1	$(4,1)$	56	4	3
8	$(4,2)$	296	12	8
9	$\mathbf{(4,3)}$	330	13	9
20	$(4,3)$	758	26	16
21	$\mathbf{(4,4)}$	782	29	19

Appendix

Table 3: The naive policy of minimizing

w_{\max}

applied to

I[1,0,-\!1,1,2,0,0]

. Rows show the first and last step of each

w_{\max}

plateau. Bold entries mark forced escalations. After 50 steps, both the maximum weight and the number of sector non-master integrals have grown significantly.

Step	$w_{\max}$	Valid	Non-masters
		actions	all	sector
0	$(5,1)$	56	4	2
2	$(4,1)$	151	4	3
10	$(4,1)$	488	3	3
11	$(3,0)$	556	1	1
14	$\mathbf{(5,0)}$	644	3	1

Appendix

Table 4: Alternative naive policy (minimize sector non-masters) applied to

I[1,0,-\!1,1,2,0,0]

. The sector count stays at 1 from Step 11 onward, but

w_{\max}

escalates without bound: the naive policy repeatedly applies

\partial_{k_{1}}\!\cdot\!k_{1}

to raise the highest propagator index by 1, creating an infinite staircase in

r

$r$	$s$	Kira	Sailir	Kira	Sailir	Mem. ratio	Jobs	Cache hits	Steps
		Time (s)		Memory (MB)			Sailir Details
10	4	8.4	601	159	2801	17.6	3815	6142	18774
11	4	14.5	658	265	3016	11.4	6426	11120	28780
12	4	24.2	1061	418	2980	7.1	7806	15497	37684
13	4	36.0	1508	640	2689	4.2	9096	17227	41321
10	5	25.6	1318	350	3212	9.2	4902	9199	28316

Appendix

Table 5: Benchmark comparison of Kira vs. Sailir on 16 integrals from the two-loop triangle-box topology. Time: Kira total time vs. Sailir ideal parallel time (longest chain of sequential dependencies). Memory: Kira peak memory vs. Sailir maximum per-CPU worker memory. Mem. ratio: Sailir memory / Kira memory (values

<1

indicate Sailir uses less memory). Jobs: total one-step worker jobs submitted. Cache hits: number of times a previously computed reduction was reused. Steps: total beam search steps across all workers.

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