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Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching
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Proposes Adjoint Sampling, an on-policy SOC-based diffusion sampler that uses Reciprocal Adjoint Matching with a replay buffer to make many gradient updates per energy evaluation, and scales to LJ-55 and molecular conformer generation.
Problem
Learning diffusion samplers from unnormalized energy functions is expensive because existing SOC/adjoint methods need SDE simulation and many energy evaluations per gradient update, which prevents scaling to expensive energy models such as molecular DFT-level energies.
Approach
Reformulate Adjoint Matching for zero-drift base processes so the lean adjoint is analytic (the terminal energy gradient), then sample intermediate states X_t from the closed-form base bridge conditioned on X_1 (Reciprocal Adjoint Matching), and reuse a replay buffer of (X_1, grad g) pairs for many regression updates.
Key results
- On LJ-55, Adjoint Sampling reaches E(·) W2 of 58.04 versus 93.53 for iDEM, using 0.002 energy evaluations per gradient update versus 512 for iDEM.
- On DW-4, Adjoint Sampling attains path-ESS 0.627, higher than PIS (0.462) and DDS (0.461).
- On SPICE with relaxation, Cartesian Adjoint Sampling (+pretrain) reaches 96.65 recall coverage versus 81.61 for RDKit ETKDG.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Reciprocal projection instead of storing trajectory sample pairs (ablation) | Adjoint Sampling · LJ-13 | E(·) W2: 2.91 ±1.39 -> 2.40 ±1.25 | Helped |
| use Reciprocal projection instead of storing trajectory sample pairs (ablation) | Adjoint Sampling · DW-4 | W2: 0.63 ±0.11 -> 0.62 ±0.06 | Helped |
| use Reciprocal projection instead of storing trajectory sample pairs (ablation) | Adjoint Sampling · LJ-55 | path-ESS: 0.094 ±0.025 -> 0.066 ±0.037 | Hurt |
| use torsion-angle rather than Cartesian coordinates | Adjoint Sampling · GEOM-DRUGS | Recall coverage: 60.93 ±35.15 -> 72.91 ±31.17 | Helped |
| add Bridge Matching pretraining on RDKit samples | Cartesian AdjSampling · GEOM-DRUGS | Recall coverage: 60.93 ±35.15 -> 72.98 ±30.82 | Helped |
| apply relaxation after sampling | Cartesian AdjSampling (+pretrain) · GEOM-DRUGS | Recall coverage: 72.98 ±30.82 -> 87.01 ±22.79 | Helped |
| use Torsional AdjSampling instead of RDKit ETKDG | Conformer generation · GEOM-DRUGS | Recall coverage: 63.51 ±34.74 -> 72.91 ±31.17 | Helped |
| use Cartesian AdjSampling instead of RDKit ETKDG | Conformer generation · GEOM-DRUGS | Precision coverage: 69.77 ±38.23 -> 28.44 ±27.77 | Hurt |
| use Adjoint Sampling instead of iDEM | Synthetic energy sampling · LJ-55 | E(·) W2: 93.53 ±16.31 -> 58.04 ±20.98 | Helped |
Surprising findings
- Adjoint Sampling w/o reciprocal projection performs worse than the full method on most metrics, highlighting the importance of the reciprocal projection.
- Single-MC-sample iDEM is unstable, with much worse metrics than the full iDEM.
Limitations
- Precision on GEOM-DRUGS is lower than RDKit ETKDG for Adjoint Sampling variants.
- Cartesian sampling needs an additional bond-length regularizer to target the correct molecule.
Experimental setup
Datasets: SPICE GEOM-DRUGS
Benchmarks: DW-4 LJ-13 LJ-55 SPICE conformer recall/precision GEOM-DRUGS conformer recall/precision
Metrics: W2 E W2 path-ESS coverage recall coverage precision AMR
Baselines: PIS DDS LogVariance iDEM RDKit ETKDG Adjoint Sampling w/o RP (ablation)
Builds on: Adjoint Matching Path Integral Sampler Denoising Diffusion Sampler iDEM Bridge Matching
Closest papers by meaning
- Non-equilibrium Annealed Adjoint Sampler (NeurIPS 2025 · similarity 0.93)
- Adjoint Schrödinger Bridge Sampler (NeurIPS 2025 · similarity 0.92)
- AdjointDEIS: Efficient Gradients for Diffusion Models (NeurIPS 2024 · similarity 0.91)
- On scalable and efficient training of diffusion samplers (NeurIPS 2025 · similarity 0.91)
- Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models (NeurIPS 2025 · similarity 0.89)
- Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics (ICLR 2025 · similarity 0.88)
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