π-Distill: QCD & PDE-Constrained Diffusion

• π-Distill is a dual framework that applies low-rank projection for high-precision lattice QCD and physics-informed diffusion model distillation.
• In lattice QCD, it employs Laplacian eigenmode projection to create distilled quark fields, enhancing multi-hadron spectroscopy and energy extraction via variational analysis.
• In physics-inspired machine learning, the PIDDM approach distills a teacher diffusion model into a single-step generator by enforcing PDE residuals for accurate system modeling.

π-Distill refers to two distinct methodologies developed for (1) high-precision correlation measurements in lattice quantum chromodynamics (QCD), and (2) the efficient, physically-constrained distillation of score-based diffusion models for generative modeling of systems governed by partial differential equations (PDEs). Both domains exploit a common principle: low-rank or post-hoc projection in order to enforce structural or physical constraints, either on quantum field correlators or on neural generative models.

1. Distillation for Lattice QCD: The π-Distill/Distillation Framework

The distillation method in lattice QCD constructs low-rank projectors onto the subspace of low-lying eigenmodes of the gauge-covariant Laplacian on each time-slice. These projectors are used to define smeared (“distilled”) quark fields, yielding all-to-all quark propagators (or “perambulators”) with significantly improved operator overlap and statistical quality for multi-hadron spectroscopy.

Distillation Operator Construction

On each time-slice $t$, the three-dimensional gauge-covariant Laplacian $-\nabla^2_{xy}(t)$ is diagonalized: $$-\nabla^2_{xy}(t)\,\xi^{(k)}_y(t) = \lambda_k(t)\,\xi^{(k)}_x(t)$$ Ordering the eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots$, the lowest $N$ eigenmodes define the distillation space: $V(t) = [\xi^{(1)}(t), \dots, \xi^{(N)}(t)]$ The rank-$N$ distillation projector is

$P(t) = V(t) V(t)^\dagger$

The smeared (distilled) quark field is then given by

$\psi_s(x,t) = [P(t)\psi(\cdot, t)]_x$

All creation and annihilation operators for composite hadrons are built from $\psi_s$; the corresponding propagators form perambulators: $$\tau_{kl}(t,0) = \xi^{(k)\,\dagger}(t) D^{-1}(t,0) \xi^{(l)}(0)$$ where $D$ is the Dirac operator (Woss et al., 2016, Lachini et al., 2021).

2. Extraction of Energy Spectra and Scattering Information

π-Distill provides the foundation for robust variational analysis using large operator bases. The correlator matrix

$C_{ij}(t) = \langle O_i(t) O_j^\dagger(0) \rangle$

is constructed using interpolating operators projected onto definite lattice momenta and irreducible representations. A generalised eigenvalue problem (GEVP) is solved: $C(t) v_n(t) = \lambda_n(t, t_0) C(t_0) v_n(t)$ with principal correlators $\lambda_n(t, t_0)$ yielding the energies through multi-exponential fits.

Optimised “single-pion” and two-meson operators are constructed using GEVPs within each irrep, maximising overlap with target states. Combining these with Lüscher’s finite-volume method enables extraction of scattering phase shifts, particularly in channels dominated by elastic two-pion ($\pi\pi$) states, e.g., for the extraction of the ρ resonance properties (Woss et al., 2016).

3. Implementation Choices: Rank Dependence, Stochastic and Exact Distillation

The choice of distillation rank ($N$ or $N_{\rm ev}$) controls both smearing radius and computational cost. Typically, $32 \leq N \leq 384$ (e.g., $N=128$ on $32^3$ lattices) or $N_{\rm ev}=64$ on a $48^3\times96$ lattice is used. The smearing profile

$$\Psi(r) = \sum_{x,t} \sqrt{ \operatorname{tr}[ S_{x, x+r}(t) S_{x+r, x}(t) ] }$$

shows rapid saturation for $N_{\rm ev} \gtrsim 60$, with diminishing returns at higher ranks.

Two main approaches exist:

• Exact distillation: Formation of all perambulators in the low-mode subspace, providing optimal error at moderate inversion cost ($4N_{\rm ev}$ per source).
• Stochastic distillation: Uses noise in the LapH subspace with dilution schemes, reducing the number of inversions but introducing stochastic noise, which requires normalization by cost to compare accuracy.

Empirically, exact distillation at $N_{\rm ev}=64$ yields optimal performance for pion and $K\pi$ correlation functions at physical masses, with consistent energies and efficient momentum projection. Stochastic distillation is suboptimal for this cost regime (Lachini et al., 2021).

4. Numerical Parameters, Precision, and Recommended Practices

Key implementation and performance characteristics established in the literature are summarized below.

Lattice Volume / Parameters	π-Distill Key Choices	Outcome / Recommendation
$32^3\times256$, $a_s\approx 0.12\,$fm, $m_\pi=236$ MeV	$N=128$ (4–10% spatial dim.), 20–35 $\bar qq$+5–6 $\pi\pi$ ops/irrep	Energies/phase shifts converge at few-% level for $N\geq128$ (Woss et al., 2016)
$48^3\times96$, $a\simeq0.114$ fm, $m_\pi=139$ MeV	$N_{\rm ev}=64$, exact distillation	Best cost–error trade-off; effective for large bases (Lachini et al., 2021)

Optimal practice is to select $N$ ($N_{\rm ev}$) such that the smearing radius covers the physical pion's size but avoids excessive computation. For the $\pi\pi$ channel up to the ρ resonance, convergence is achieved for ranks $\gtrsim128$ on moderate volumes; at the physical point, $N_{\rm ev}=64$ suffices.

5. Physics-Informed Distillation of Diffusion Models (PIDDM)

In generative modeling of physics-governed systems, π-Distill (“PIDDM”) refers to the post-hoc distillation of pretrained score-based diffusion models, with explicit enforcement of PDE residual constraints at the level of the one-step student model output. The method builds upon a teacher diffusion model (trained via score-matching on noisy data) and distills its iterative sampler into a single neural network, training with a combined regression-to-teacher and physics residual penalty: $$\mathcal L_{\rm distill}(\theta') = \mathcal L_{\rm sample} + \lambda_{\rm train} \mathcal L_{\rm PDE}$$ where

$$\mathcal L_{\rm PDE} = \mathbb E_{\epsilon} \| \mathcal R(d_{\theta'}(\epsilon)) \|^2, \quad \mathcal R(x) = [\mathcal F[u], \mathcal B[a]]^\top$$

with physical constraints $\mathcal F, \mathcal B$ specified for the PDE class (Zhang et al., 28 May 2025).

Unlike earlier approaches, which enforced constraints on the score-model’s posterior mean (subject to Jensen's gap), PIDDM eliminates this gap by applying penalties solely to actual (fully generated) clean samples. This supports strict PDE satisfaction for forward, inverse, and partial observation generation.

6. Algorithmic Procedure, Experimental Validation, and Limitations

PIDDM operates in two stages:

• Distillation: Generate a (noise, teacher sample) dataset, train the student to regress to teacher output plus minimize PDE residual.
• Inference: For constrained tasks, refinement iterations may further penalize PDE violation, though single-step generation already achieves competitive accuracy.

Validation over benchmarks (Darcy flow, Poisson, Burgers’, etc.) demonstrates that PIDDM achieves significantly reduced PDE error ($\sim 0.15-0.2\times10^{-4}$, NFE as low as 1) compared to diffusion models with stepwise enforcement. Ablations indicate that higher teacher fidelity and moderate loss tradeoffs optimize sample quality and physics accuracy. The method is limited by reliance on a well-trained teacher and a discretized PDE operator. Future directions include embedding constraints at the architectural/tokenizer level, handling stiffer equations, and scaling to 3D/multiphysics scenarios (Zhang et al., 28 May 2025).

7. Significance and Broader Context

Distillation-based frameworks, as exemplified by π-Distill, enable efficient, low-noise extraction of physically meaningful observables—whether in precision lattice spectroscopy (via Laplacian eigen-projection and variational analysis) or fast, physical constraint-satisfying generative modeling (via post-hoc one-step network distillation). For lattice QCD, the method is now a standard tool for large-volume, high-precision studies of multi-hadron systems; for physics-inspired machine learning, PIDDM provides a principled solution to enforcing hard physical constraints in generative pipelines. Both applications illustrate the general efficacy of low-rank or sample-level projection for balancing fidelity, tractability, and structure in data-driven modeling.