# PDE Sinkhorn with asymmetric geometric boundaries

- ID: 1094
- Canonical URL: https://synthcore.org/idea/1094/pde-sinkhorn-with-asymmetric-geometric-boundaries
- API JSON: https://synthcore.org/api/idea/1094.json
- API Markdown: https://synthcore.org/api/idea/1094.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.17904](https://arxiv.org/abs/2607.17904)
- Category: sampling
- Solves: accuracy, stability, sample-efficiency
- ML areas: diffusion-sampling, diffusion, loss
- Math tags: pde, optimal-transport, stochastic-processes, numerical-analysis
- Ratings: usefulness 7/10; difficulty 7/10; novelty 8/10

## Idea description

Build a Schrödinger-bridge solver that represents the two Sinkhorn scaling factors as solutions of forward and backward Kolmogorov PDEs, rather than requiring explicit transition-density evaluation. Enforce an oblique Neumann condition on the backward factor and a normal no-flux condition on the forward factor, allowing degenerate diffusion and hard domain boundaries to be handled directly.

## Mathematical statement

Let \(g_i(x,t)\) be the admissible diffusion fields, \(G(x,t)=\sum_i g_i g_i^\top\) their Gram matrix, and \(b(x,t)\) the drift. The backward generator is \(\mathcal L_tu=b\cdot\nabla u+\tfrac12\sum_i(g_i\cdot\nabla)^2u\). With \(n=\nabla\xi/\|\nabla\xi\|\) and \(r=Gn/(n^\top Gn)\), the backward Schrödinger factor \(\varphi_t\) satisfies \(\partial_t\varphi_t+\mathcal L_t\varphi_t=0\) and \(r\cdot\nabla\varphi_t=0\) on the boundary. The forward factor \(\hat\varphi_t\) evolves under the adjoint Fokker-Planck operator and uses normal no-flux, \(n\cdot J_t[\hat\varphi_t]=0\), where \(J_t[q]=bq-\tfrac12\sum_i g_i(g_i\cdot\nabla q)\) is the probability current in the directional-diffusion approximation. The bridge density factorizes as \(\rho_t=\varphi_t\hat\varphi_t\). Sinkhorn alternates endpoint rescaling of the factors to match prescribed densities \(\rho_0\) and \(\rho_1\). The transferable result is that boundary conditions are imposed directly on PDE factors and fluxes, avoiding an explicit transition kernel.

## Key formulas

- $$\mathcal L_tu=b(x,t)\cdot\nabla u+\frac12\sum_{i=1}^m\bigl(g_i(x,t)\cdot\nabla\bigr)^2u,$$
- $$\partial_t\varphi_t+\mathcal L_t\varphi_t=0,\qquad r(x,t)\cdot\nabla\varphi_t=0\quad\text{on }\partial\mathcal X,$$
- $$\partial_t\hat\varphi_t=\mathcal L_t^*\hat\varphi_t,\qquad n(x)\cdot J_t[\hat\varphi_t](x)=0\quad\text{on }\partial\mathcal X,$$
- $$\varphi_0\leftarrow\varphi_0\frac{\rho_0}{\varphi_0\hat\varphi_0},\qquad \hat\varphi_1\leftarrow\hat\varphi_1\frac{\rho_1}{\varphi_1\hat\varphi_1},\qquad \rho_t=\varphi_t\hat\varphi_t.$$

## Implementation notes

Use this as a low-dimensional constrained generative-model or bridge module where PDE discretization is feasible. Create a mesh or finite-volume grid covering \(\mathcal X\), store positive arrays for \(\varphi_t\) and \(\hat\varphi_t\), and define \(\xi\), \(n\), \(g_i\), and \(b\) at cell centers and boundary faces. Discretize \(\mathcal L\) with directional derivatives along each \(g_i\), using an implicit or positivity-preserving scheme. During every backward solve, impose \(r\cdot\nabla\varphi=0\) by eliminating the boundary ghost value along \(r=Gn/(n^\top Gn+\varepsilon)\). During every forward solve, impose zero normal numerical flux by setting the boundary-face current to zero rather than reflecting particles individually. Run iterative proportional fitting in log-space: initialize \(\varphi_0=1\), propagate it forward, rescale the terminal factor to match \(\rho_1\), propagate the adjoint factor backward, rescale the initial factor to match \(\rho_0\), and repeat with damping \(q\leftarrow(1-\eta)q+\eta q_{\rm new}\). The PDE equations, factorization, and asymmetric boundary conditions are computed from the paper; grid resolution, time-step stability, and damping are empirical. First test a 2D annulus with two separated Gaussian endpoint densities and compare against kernel Sinkhorn, unconstrained diffusion, and particle rejection. Track marginal KL error, boundary mass leakage, bridge action, and iteration count. Success means stable convergence with near-zero boundary flux and lower endpoint error than rejection-based particle SB under anisotropic or rank-deficient diffusion.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built a NumPy/SciPy MVP containing the oblique boundary calculation, conservative finite-volume reflected diffusion, explicit stability sweep, and endpoint Sinkhorn scaling comparison. The mechanism manifested: for G=[[1,c],[c,1]], the oblique residual stayed 0 while the normal-BC residual was exactly c; no-flux propagation conserved mass to 1e-12 or better across diffusion strengths; and positivity held through ratio 1.0 of the predicted dt=7.8125e-5 bound but failed strongly at 1.05. Reflected and unconstrained kernel Sinkhorn both matched endpoint marginals to roughly 1e-15 KL in this easy 1D test, so no accuracy win over the baseline was demonstrated.

### Mechanism check

- Verdict: Built a NumPy/SciPy MVP containing the oblique boundary calculation, conservative finite-volume reflected diffusion, explicit stability sweep, and endpoint Sinkhorn scaling comparison. The mechanism manifested: for G=[[1,c],[c,1]], the oblique residual stayed 0 while the normal-BC residual was exactly c; no-flux propagation conserved mass to 1e-12 or better across diffusion strengths; and positivity held through ratio 1.0 of the predicted dt=7.8125e-5 bound but failed strongly at 1.05. Reflected and unconstrained kernel Sinkhorn both matched endpoint marginals to roughly 1e-15 KL in this easy 1D test, so no accuracy win over the baseline was demonstrated.
- Confidence: 8/10
- Limitations: This is a low-dimensional numerical MVP, not the requested 2D annulus experiment. The backward oblique condition is verified algebraically and is not yet integrated into a full 2D backward-factor PDE solve; the forward test uses pure diffusion, no drift, and the Sinkhorn comparison uses a precomputed matrix exponential rather than repeated time-stepped factor propagation. No particle-rejection baseline, bridge action, runtime benchmark, or rank-deficient multidimensional directional diffusion was tested.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0152
- Paired wins: 1/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/437/bench_report.json)
- [experiment.py](https://synthcore.org/code/437/experiment.py)
- [pde_boundary_track.py](https://synthcore.org/code/437/pde_boundary_track.py)
- [report.md](https://synthcore.org/code/437/report.md)
- [report_bench_2026-08-31T121148.md](https://synthcore.org/code/437/report_bench_2026-08-31T121148.md)
- [results.json](https://synthcore.org/code/437/results.json)
- [run_bench.py](https://synthcore.org/code/437/run_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/437)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.