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 μbase(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 μθ(x) that approximates the true underlying real-world distribution ν(x) .
Figure 2: ADA finds the closest distribution μθ∗ to a base distribution μbase that satisfies the observable constraints o#(i)μθ=o#(i)ν for all observables i∈{1,…,m} . The resulting feasible set is denoted by Mo(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 fϕ(i) learns the observable Wasserstein distance to align the model with the distributional observable constraint.
ALGORITHM 1: Adversarial Distribution Alignment from a Pretrained Generative Model
Method
Match Expectation
Match Distribution
Requires Training
Single
Multiple
Single
Multiple
Conditional Generative Model
✓
✗
✓
✗
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 △ ), 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 ℓ1 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 W1↓
Held-out FES JSD ↓
Constraint
MD
Rg
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 ( Rg ), 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.
W1 to Experiment ( ↓ )
Max RMSD
MeanDist
Rg
(nm, ↓ )
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 {(±3,±3,±3)} Å. The base distribution is uniform, while the target distribution has heterogeneous variances and mixture weights.
Method
Energy W2 ↓
Cluster KL ↓
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.
Simulation and experimental measurements provide complementary data for learning spatiotemporal physical systems, but standard simulation-to-experiment fine-tuning optimizes only the experimental objective after transfer and can degrade simulation performance. We formulate simulation--experiment prediction as a multi-objective learning problem with domain-specific simulation and experimental risks. On four fluid systems from RealPDEBench and two model capacities, we compare Simulation only, Experiment only, Sim→Exp, and Joint training, evaluating every final model on both held-out domains. Sim→Exp tends to specialize more strongly to experimental data at the cost of simulation-domain forgetting. Joint training consistently achieves the best balanced performance over a broad range of simulation--experiment evaluation weightings, while substantially improving simulation retention over Sim→Exp. Joint also better preserves simulation-only fields absent from experimental measurements. Project page: https://mahindrautela.github.io/morph.
Mahindra Rautela, Alexander Scheinker, Ayan Biswas +3
Domain adaptation is widely used to make neural networks trained on simulations applicable to experimental data. Its premise is that the two domains differ only in nuisances, and that the quantity of interest is distributed identically in both. In physics neither assumption holds: simulations can be wrong about the physics, and the distribution of the target quantity - an energy spectrum, a redshift distribution - is often the measurement itself. We study the consequences of such mismatches on a toy air-shower benchmark in which a detector-response nuisance, a physical simulation shift, and an energy-spectrum shift can be switched on separately or together. Standard adversarial adaptation handles the conditional shifts, but once the two spectra differ it aligns them, replacing an uncontrolled bias by one anchored on the simulation prior. We present adaptive domain adaptation, which reweights the simulated events so as to focus domain adaptation on the genuine physical mismatch alone. Since the predicted spectrum depends on model training configuration, we provide a label-free model selection rule for selecting the near-the-best operation point.
Ivan Kharuk
Institute for Nuclear Research of the Russian Academy of Sciences · Moscow Institute of Physics and Technology · Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow, 117312, Russia +1
Physics-informed generative models augment a data-fitting objective with residual penalties that encourage physical constraints to hold in generated samples. However, improving constraint satisfaction need not improve fidelity to the data distribution. We study this conflict in residual-regularized diffusion models and identify two possible consequences. First, the same prediction error can incur different residual costs near different feasible states, preferring states with lower residual sensitivity (avoidance). Second, this preference can extend to feasible states outside the data support, where the residual also vanishes (leakage). Motivated by this analysis, we propose a data-anchored penalty that weights prediction errors by the residual Jacobian at the clean training sample and normalizes the resulting magnitude. This removes the incentive to relocate to less sensitive states without changing the model architecture or sampling procedure. Experiments on a controlled synthetic system, three PDE systems, and molecular transition-state generation with flow matching demonstrate the benefits of anchoring. Code is available at https://github.com/TanjaBien/anchored_diffusion.