Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions

Kehan Wang

cs.LGOct 27, 2026

Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions

Authors: Kehan Wang

Organizations: School of Data Science, The Chinese University of Hong Kong, Shenzhen, China

Abstract

Stationary tail computation for high-dimensional reflected Brownian motions (RBMs) remains computationally challenging. Building on NN4MGF's shared-coordinate basic-adjoint-relationship (BAR) framework, we develop Holomorphic Cauchy Networks (HCNs). Native complex-valued rational Cauchy subnetworks enforce holomorphicity on pole-free domains, removing sampled Cauchy-Riemann penalties and their input-derivative graphs. Exact origin anchoring and boundary-coordinate exclusion enforce structural identities, while anchored interactions represent dependent transforms. Under holomorphic-log assumptions, we establish simultaneous value and first-derivative approximation for the constrained architectures and a mixed-derivative lower bound for coordinate-only representations under nonzero cross-covariance. Closed-form real-axis derivatives and sequential backpropagation further control memory. On the same NVIDIA H100 NVL, the tested 30D HCN configuration uses approximately tenfold less peak allocated memory and has 9.7-fold lower mean update time than NN4MGF; extrapolation to the respective scheduled update counts gives a projected full-schedule update-time ratio of 58.2. With 64 million sampled base inputs per high-dimensional run, HCN reaches 50 dimensions on a laptop GPU with 6.367% mean relative error over thresholds 1 through 8. Degree-matched comparisons yield lower reported mean relative errors over the primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark. An exploratory 10D correlated example illustrates the interaction extension.

Figures & tables

Benchmark	Updates ( $U$ )	$B_{\rm BAR}$	$B_{\rm bd}$	$B_{\rm real}$	Base inputs ( $N_{\rm train}$ )
2D, both models	15,000	3,416	427	427	64.05M
10D non-product	25,000	2,048	256	256	64M
20D product	25,000	2,048	256	256	64M
30D product	50,000	1,024	128	128	64M
50D product	100,000	512	64	64	64M

Table 1: HCN batches and fixed training endpoints.

Quantity	HCN	NN4MGF
Real scalar parameters	17,028	43,080
Base inputs / update	1,280	32,768
Measured updates (after 10 warmup)	200	200
Time for 200 updates (s)	33.023	320.209
Mean time / update (s)	0.165113	1.601046
Peak allocated memory (GiB)	7.805	77.815

Table 2: Thirty-dimensional training profiles on the same NVIDIA H100 NVL.

Benchmark	Method	$M$	$1{:}5$	$1{:}8$	$1{:}10$
2D product	HCN	5	$2.147$	$2.213$	$9.737$
2D product	NN4MGF (published)	5	$2.448$	$8.050$	$13.915$
2D Harrison	HCN	5	$1.422$	—	—
2D Harrison	NN4MGF (published)	5	$6.684$	—	—
20D product	HCN	8	$1.441\pm 1.322$	$7.632\pm 1.565$	$12.264\pm 3.678$
20D product	NN4MGF (archived)	8	$2.802$	$8.148$	$18.908$

Table 3: Tail accuracy at fixed training endpoints (MRE, percent).

Figure 1: 2D tails at Degree 5. HCN uses the fixed 15,000-update endpoints; NN4MGF uses published values from Dai and Zhang (2026) . References are analytical in (a) and density-based in (b).

Figure 2: Product-form tails at Degree 8. Panel (a) shows the three 20D HCN runs and the archived NN4MGF checkpoint. Panels (b) and (c) show the fixed endpoints at 30D and 50D.

	$t=1{:}5$		$t=1{:}8$		$t=1{:}10$
Dimension	$M=5$	$M=8$	$M=5$	$M=8$	$M=5$	$M=8$
20D	3.391	0.073	4.044	0.122	9.430	0.332
30D	3.218	0.152	3.955	0.178	8.397	0.375
50D	34.866	0.507	23.898	0.414	23.352	0.563

Table 4: Fixed-Talbot inversion of the exact product-form transform: Degree 5 versus Degree 8 (MRE, percent).

Seed	Cauchy MRE (%)	Complex-tanh MRE (%)
12345	9.003	32.990
22345	7.966	19.823
32345	5.927	13.502
Mean $\pm$ SD	$7.632\pm 1.565$	$22.105\pm 9.942$

Table 5: Activation comparison at 20D and Degree 8: MRE on

1{:}8

(percent).

Update	$1{:}8$ MRE (%)	Role
92,000	6.650	Diagnostic checkpoint
94,000	5.299	Diagnostic checkpoint
96,000	6.845	Diagnostic checkpoint
98,000	7.155	Diagnostic checkpoint
100,000	6.367	Reported endpoint

Table 6: Late-training tail accuracy along the 50D trajectory.

Figure 3: Exploratory 10D non-product tails at Degree 8: HCN interaction model compared with the pilot simulation reference and independent-marginal tail.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Model	$q_{\rm int}$	$q_{\rm bd}$	$g_{\rm int}$	$g_{\rm bd}$	Parameters
2D product	128	128	—	—	15,236
2D Harrison	95	128	109	—	15,236
10D non-product	95	128	109	109	23,925
20D product, Cauchy	128	128	—	—	16,388
20D product, tanh	128	128	—	—	15,108
30D product	128	128	—	—	17,028

Appendix

Table A.1: Hidden widths of the scalar subnetworks and real scalar parameter counts.

Setting	Value
Arithmetic	complex128 network values and weights; float64 real quantities.
Embeddings	$p_{e}=p_{b}=16$ complex components per coordinate and boundary face, initialized $\mathcal{N}(0,0.01^{2})$ .
Cauchy initialization	$a_{r}=b_{r}=0.1$ , $s_{r}=0.5$ , $s_{\min}=10^{-3}$ ; $\rho_{r}=\log(\exp(0.5-s_{\min})-1)$ .
Optimizer	Adam, learning rate $5\times 10^{-5}$ for the first 80% of updates, $10^{-5}$ thereafter.
Loss and gradients	Unit weights $(1,1,1,1)$ in ( 27 ); global accumulated gradient norm clipped at $1.0$ .
Pair chunk cap	32,768 scalar inputs in 2D and product-form models; 4,096 in 10D.

Appendix

Table B.1: Implementation hyperparameters and numerical settings.

Benchmark	Checked object	Tolerance	Maximum error
2D product	Coordinate block	$2\times 10^{-7}$	$1.131\times 10^{-9}$
2D Harrison	Coordinate block	$2\times 10^{-7}$	$2.99\times 10^{-9}$
2D Harrison	Interaction block	$2\times 10^{-7}$	$3.577\times 10^{-10}$
10D non-product	Full log-transform vector	$1\times 10^{-5}$	$1.818\times 10^{-10}$

Appendix

Table B.2: Finite-difference Jacobian verification checks at initialization.

Model	$M$	$t$	$r_{\rm inv}$	$r_{\rm net}$	$r_{\rm total}$
NN4MGF	5	8	-2.516	+18.374	+15.858
HCN (32345)	5	8	-2.516	-16.412	-18.928
NN4MGF	8	8	-0.447	-28.582	-29.028
HCN (32345)	8	8	-0.447	-8.037	-8.484

Appendix

Table C.1: Signed relative-error decomposition at 20D (

t=8

, percent).

Model	$M$	$1{:}5$	$1{:}8$	$1{:}10$
20D HCN (3-seed mean)	5	2.983	7.730	19.347
20D HCN (3-seed mean)	8	1.441	7.632	12.264
20D NN4MGF	5	4.013	8.006	16.801
20D NN4MGF	8	2.802	8.148	18.908
30D HCN	5	3.429	10.107	19.084
30D HCN	8	1.683	6.546	8.872

Appendix

Table D.1: Fixed-endpoint MREs under Degrees 5 and 8 (percent).

Update	$1{:}5$	$1{:}8$	Role
92,000	1.524	6.650	Diagnostic checkpoint
94,000	1.611	5.299	Diagnostic checkpoint
96,000	1.318	6.845	Diagnostic checkpoint
98,000	1.473	7.155	Diagnostic checkpoint
100,000	1.495	6.367	Reported endpoint

Appendix

Table D.2: Late-training checkpoints along the 50D trajectory at Degree 8 (MRE, percent).

$t$	Simulation ref.	HCN interaction	Independence	HCN RE (%)	Indep. RE (%)
1	0.972040	1.000000	0.989814	2.88	1.83
2	0.720910	0.722273	0.766398	0.19	6.31
3	0.400300	0.398938	0.399920	0.34	0.10
4	0.191240	0.187112	0.168775	2.16	11.75
5	0.084430	0.086949	0.065373	2.98	22.57
6	0.035760	0.039650	0.024513	10.88	31.45

Appendix

Table D.3: Ten-dimensional correlated non-product RBM at Degree 8.

Task	Python entry point
Verify SHA-256 hashes and summaries	audit/audit_results.py
Rebuild all main and supplement tables/figures	build_results.py , then scripts/build_appendix_tables.py
Compile main and supplement PDFs	scripts/build_pdfs.py

Appendix

Table E.1: Entry points for verifying records and rebuilding all manuscript tables, figures, and PDFs.

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The stationary distribution of reflected Brownian motion (RBM) plays an important role in the analysis of high-dimensional stochastic systems, yet closed-form solutions are known only for a few special cases. Computing important performance metrics, such as tail probabilities, is even more intractable, despite their practical relevance. In this paper, we develop a deep learning approach that accurately and efficiently learns the Laplace transform of high-dimensional RBMs based on the basic adjoint relationship (BAR). Our framework combines a careful design of the loss function, training data sampling procedure, and neural network architecture. We evaluate the proposed method on RBM instances with known ground-truth tail probabilities and demonstrate near-perfect prediction in high-dimensional settings, highlighting its potential as a general tool for analyzing stochastic systems beyond analytically tractable regimes. Our code can be found at https://github.com/zhangz73/NN4MGF.

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