Bridging the Simulation-to-Experiment Gap with Generative Models using Adversarial Distribution Alignment

Kai Nelson; Tobias Kreiman; Sergey Levine; Aditi S. Krishnapriyan

cs.LGApr 1, 2026

Bridging the Simulation-to-Experiment Gap with Generative Models using Adversarial Distribution Alignment

Authors: Kai Nelson, Tobias Kreiman, Sergey Levine, Aditi S. Krishnapriyan

Organizations: University of California, Berkeley · Lawrence Berkeley National Laboratory (LBNL)

Abstract

A fundamental challenge in science and engineering is the simulation-to-experiment gap. While we often possess prior knowledge of physical laws, these physical laws can be too difficult to solve exactly for complex systems. Such systems are commonly modeled using simulators, which impose computational approximations. Meanwhile, experimental measurements more faithfully represent the real world, but experimental data typically consists of observations that only partially reflect the system's full underlying state. We propose a data-driven distribution alignment framework that bridges this simulation-to-experiment gap by pre-training a generative model on fully observed (but imperfect) simulation data, then aligning it with partial (but real) observations of experimental data. While our method is domain-agnostic, we ground our approach in the physical sciences by introducing Adversarial Distribution Alignment (ADA). This method aligns a generative model of atomic positions -- initially trained on a simulated Boltzmann distribution -- with the distribution of experimental observations. We prove that our method recovers the target observable distribution, even with multiple, potentially correlated observables. We also empirically validate our framework on synthetic, molecular, and experimental protein data, demonstrating that it can align generative models with diverse observables. Our code is available at https://kaityrusnelson.com/ada/.

Figures & tables

Figure 1: ADA aligns generative models trained on approximate simulation data with the real world by leveraging partial experimental observations. (1) ADA starts from a base generative model $\mu_{\text{base}}(x)$ trained on simulation data, such as molecular dynamics simulations using a classical force field. (2) ADA aligns the base model with multiple, potentially correlated partial experimental observations (e.g., radial distribution functions (RDFs) or nuclear magnetic resonance (NMR) measurements). (3) ADA yields an aligned model $\mu_{\theta}(x)$ that approximates the true underlying real-world distribution $\nu(x)$ .

Figure 2: ADA finds the closest distribution $\mu_{\theta}^{*}$ to a base distribution $\mu_{\text{base}}$ that satisfies the observable constraints $o^{(i)}_{\#}\mu_{\theta}=o^{(i)}_{\#}\nu$ for all observables $i\in\{1,\dots,m\}$ . The resulting feasible set is denoted by $\mathcal{M}_{o}(\mathcal{X})$ (shown in orange). The KL term in Equation 2 acts as an entropic regularizer, reflecting the fact that the observable constraints in Equation 1 do not uniquely specify the underlying distribution, since many states can map to the same observable values. The discriminator $f^{(i)}_{\phi}$ learns the observable Wasserstein distance to align the model with the distributional observable constraint.

(o^{(i)})_{\#}\mu_{\theta}\approx\frac{1}{N}\sum_{j=1}^{N}\delta_{u^{(i)}_{j}},\qquad(o^{(i)})_{\#}\nu\approx\frac{1}{N}\sum_{j=1}^{N}\delta_{\tilde{o}^{(i)}_{j}}

ALGORITHM 1: Adversarial Distribution Alignment from a Pretrained Generative Model

Method	Match Expectation		Match Distribution		Requires Training
	Single	Multiple	Single	Multiple
Conditional Generative Model	$\checkmark$	✗	$\checkmark$	✗	classifier (or CFG)
Expectation Alignment	✓	✓	✗	✗	generator
ADA (ours)	✓	✓	✓	✓	generator + discriminator

Table 1: Observable Alignment naturally handles multiple observables. Our method learns the marginal distribution of multiple, potentially correlated observables directly from data via a discriminator. In contrast, guidance-based approaches typically target a single observable and offer no guarantees for aligning multiple (potentially correlated) observables. To match the distribution of a single observable, guidance would need to rely on conditioning on samples from the target observable distribution (indicated by a

\triangle

), whereas Observable Alignment learns the distribution directly from data.

Figure 3: Comparison between expectation alignment (EA) and ADA on synthetic mixture-of-Gaussians benchmark. We align a base distribution, defined as a mixture of Gaussians centered at the corners of a cube, with a target distribution where the variances and mixture weights have been perturbed. For the observables, we use pairwise coordinate projections, yielding correlated, multimodal marginals. We report the $\ell_{1}$ pdf residual between target and generated distributions before and after alignment. ADA consistently achieves larger reductions in distributional error on multidimensional observables, even when EA uses three moments. This is taken from a kernel density estimate of 2000 samples.

	Observable marginals $W_{1}\downarrow$										Held-out FES JSD $\downarrow$
Constraint	MD		$R_{g}$		Bond		Hbond		COOH		A		B		C
	Ours	EA	Ours	EA	Ours	EA	Ours	EA	Ours	EA	Ours	EA	Ours	EA	Ours	EA
Base	0.06		0.02		0.018		0.40		0.50		0.22		0.44		0.38
+ MD	0.008	0.012	0.020	0.009	0.007	0.010	0.25	0.41	0.37	0.39	0.23	0.20	0.40	0.41	0.35	0.35
+ RoG	0.009	0.012	0.010	0.009	0.010	0.011	0.30	0.41	0.43	0.48	0.20	0.19	0.42	0.40	0.35	0.34
+ Bond	0.004	0.019	0.005	0.009	0.004	0.009	0.21	0.36	0.46	0.48	0.15	0.20	0.41	0.43	0.36	0.36

Table 2: ADA improves in alignment to the target distribution when given access to an increasing number of observables. We train a base model on constant-temperature molecular dynamics simulations using the semi-empirical GFN2-xTB potential. This model is then aligned with the distribution induced by a more accurate DFT potential using structural observables. Each column reports the distance to the reference DFT observable distribution. Moving down the rows of the table adds observables to the alignment procedure, with the bottom row representing aligning with the full observable set: mean interatomic distance (MD), radius of gyration (

R_{g}

), bond lengths, hydrogen-bond distance, and COOH–ester distance. The upper-triangular part of the left side of the table represents improvements on held-out observables (highlighted in purple). We additionally report improvement on Jensen-Shannon divergences on three fully held-out free energy surfaces: (MD, Rg), (bond length, H-bond distance), and (H-bond distance, COOH–ester distance). These surfaces are not used during training (indicated by the purple highlight) and therefore measure preservation of joint molecular structure beyond the enforced observable marginals. Compared to Expectation Alignment (EA), ADA better aligns to the target distribution (indicated with bold), especially with additional observables.

Figure 4: ADA shifts the simulated distribution to align with the experimentally measured Trp-cage structures by using only noisy, high-dimensional cryo-EM images. The base generative model is trained on MD simulations using a classical force field. ADA uses the cryo-EM images as observables to align with the experimental structures from the PDB. The long MD simulations run by a classical force field sample the unfolded states more often than seen in the 2JOF PDB entry, leading to a longer tail in the radius of gyration observable.

	$W_{1}$ to Experiment ( $\downarrow$ )		Max RMSD
	MeanDist	$\mathbf{R_{g}}$	(nm, $\downarrow$ )
Trp-Cage Base	1.26	1.07	1.18
ADA (SNR 1.0)	0.17	0.21	0.91
ADA (SNR 0.1)	0.28	0.36	0.96
BBL Base	1.36	1.08	1.76
ADA (SNR 1.0)	0.21	0.18	1.38

Table 3: ADA aligns a generative model trained on simulated data from a classical force field with experimental protein structures from the PDB using, noisy, high-dimensional cryo-EM images. We pre-train a generative model on samples of fast-folding proteins obtained from molecular dynamics simulations using a classical force field [ 38 ] . Using only the noisy cryo-EM images, ADA improves alignment with the underlying experimental distribution of structures, as measured by the Wasserstein distance to other, held-out observables. The maximum Kabsch-aligned RMSD to the experimental structures also decreases, indicating better distributional alignment at level of amino acid positions.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Synthetic mixture-of-Gaussians benchmark. The base distribution (left) is an isotropic mixture-of-Gaussians with eight components located at the vertices of a cube, with uniform mixture weights. We define the target distribution (right) by perturbing the variances and mixture weights for the Gaussians (for example, larger variance on top-left to bottom-right diagonal).

Parameter	Base (Uniform)	Target (Reweighted)
Cube Side Length	6.0 Å	6.0 Å
Number of Components	8	8
Component Positions	Cube vertices	Cube vertices
Mixture Weights:
Diagonal Pair 1	0.25	0.40
Diagonal Pair 2	0.25	0.30

Appendix

Table 4: Synthetic mixture-of-Gaussians parameters. Both distributions have eight Gaussian components positioned at the cube vertices

\{(\pm 3,\pm 3,\pm 3)\}

Å. The base distribution is uniform, while the target distribution has heterogeneous variances and mixture weights.

Method	Energy W2 $\downarrow$	Cluster KL $\downarrow$
Base	1.10	0.12
ADA (ours)	0.05	0.01
EA (1st)	1.10	0.12
EA (2nd)	0.90	0.02
EA (3rd)	0.90	0.02
EA (4th)	0.70	0.05

Appendix

Table 5: Distributional alignment metrics on mixture of Gaussians. We compare alignment using ADA against EA style approaches using an observable set of 2-dimensional projection maps. Up until the fourth order, moment-based matching is unable to recover the distribution empirically.

Figure 6: Comparison of expectation alignment (EA) and ADA across all evaluated observables on the MD17 aspirin system. While EA aligns first-order statistics, ADA more accurately captures higher-order structure, yielding closer agreement with the full target observable distributions.

Figure 7: Example cryo-EM images of Trp-cage used by ADA to align with the experimental distribution of protein structures.

	Value
Hyperparameter	Synthetic	Small Molecules	Proteins
Base Model:
Layers	MLP(512, 512, 512, 512)	8	8
Heads	N/A	12	12
Hidden Dimension	N/A	384	384
Number of Euler Steps (sampling)	N/A	45	35

Appendix

Table 6: Experiment Hyperparameters. Note that discriminator steps refers to the number of gradient steps taken between each update of the generative model, and the discriminator weight refers to the weighting of the discriminator term relative to the KL term in the generative model update step.

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