Learning Macroscopic Dynamics without Reconstructing Microscopic States

Zhichao Han; Yue Zhao; Qianxiao Li

cs.LGSep 29, 2026

Learning Macroscopic Dynamics without Reconstructing Microscopic States

Authors: Zhichao Han, Yue Zhao, Qianxiao Li

Organizations: National University of Singapore

Abstract

Modeling the temporal evolution of macroscopic properties of complex systems is an important scientific task. To predict this evolution without full microscopic simulation, a common approach encodes microstates into compact latent states, learns their evolution, and reads out macroscopic predictions from the latent trajectory. These latent states are often learned through microstate reconstruction. However, with limited latent capacity, reconstruction can favor high-variance microscopic details over information needed for macroscopic prediction. Yet jointly learning latent states and their transition without reconstruction often fails to obtain latent dynamics that support accurate macroscopic prediction. We show that this failure can arise from latent scale collapse: shrinking the latent state scale reduces training loss while macroscopic evolution error remains large. Here, we propose a reconstruction-free framework to learn latent states with their dynamics for prescribed macroscopic prediction. Training alternates between updating the latent representation with the transition and next-state latent targets fixed, and updating the transition with the latent representation fixed. At inference, the trained model predicts macroscopic states recursively from an initial microstate. Our theoretical analysis characterizes reconstruction misalignment and scale collapse under joint training, and gives a sufficient condition for local convergence to correct latent dynamics for our method. Experiments on epidemic spreading on a lattice, mixing of two particle species, and polymer stretching demonstrate that the proposed method achieves substantially better macroscopic prediction over baselines.

Figures & tables

Figure 1: Overview of TAMPL. The encoder

E_{\theta_{E}}

maps microstates

{\bm{x}}_{t}

to latent states

{\bm{z}}_{t}

, which the transition model

T_{\theta_{T}}

propagates forward and the readout

R_{\theta_{R}}

maps to macroscopic predictions

\hat{\bm{y}}_{t}

. The target encoder

\bar{E}

applied to

{\bm{x}}_{t+1}

is a copy of

E_{\theta_{E}}

and the parameters in

\bar{E}

are not trainable. Optimization alternates between representation learning (yellow) and transition learning (green), with and icons indicating trainable and frozen modules, respectively.

\bar{E}

is only updated after one representation learning stage finishes. Crossed-out decoder branches emphasize that the latent representation is trained without microstate reconstruction.

Experiment	Microstate	$n$	$m$
Lattice SIRS	Discrete lattice	$10{,}000$	$3$
Binary mixing	Particle set	$1{,}536$	$2$
Polymer extension	Grayscale image	$50{,}000$	$1$

Table 1: Experiment summary.

n

and

m

are the microscopic and macroscopic dimensions, respectively.

Figure 2: SIRS macroscopic prediction by TAMPL. The left column shows the initial lattice, with infected sites in red and susceptible sites in gray. The right three columns show the predicted and ground-truth macroscopic observations.

	Lattice SIRS		Binary Mixing		Polymer Extension
Method	RMSE	MMD	RMSE	MMD	RMSE	MMD
JLD	$0.6254$ $\pm 0.2400$	$0.2263$ $\pm 0.1200$	$0.7731$ $\pm 0.0016$	$0.3415$ $\pm 0.0021$	$0.6433$ $\pm 0.0019$	$0.1817$ $\pm 0.0029$
JLD–SIG	$0.2234$ $\pm 0.0668$	$0.0412$ $\pm 0.0235$	$0.7741$ $\pm 0.0019$	$0.3427$ $\pm 0.0024$	$0.4562$ $\pm 0.1464$	$0.0980$ $\pm 0.0558$
JLD–VIC	$0.1324$ $\pm 0.0180$	$0.0158$ $\pm 0.0040$	$0.6043$ $\pm 0.0209$	$0.2438$ $\pm 0.0105$	$0.6236$ $\pm 0.0218$	$0.1696$ $\pm 0.0089$
VAMP	$0.1955$ $\pm 0.0312$	$\mathrm{N/A}$	$0.6607$ $\pm 0.0074$	$\mathrm{N/A}$	$0.4431$ $\pm 0.0644$	$\mathrm{N/A}$
JLD–Recon	$0.4406$ $\pm 0.1848$	$0.0876$ $\pm 0.0194$	$0.6772$ $\pm 0.0862$	$0.2639$ $\pm 0.0698$	$0.6517$ $\pm 0.0097$	$0.1958$ $\pm 0.0137$

Table 2: Quantitative evaluation on the SIRS, binary mixing, and polymer extension experiments. Macroscopic features are standardized using the mean and population standard deviation of training data. We set

d_{z}=16

for SIRS,

d_{z}=8

for mixing, and

d_{z}=4

for polymer. Values are the mean

\pm

standard deviation over three independent runs. Lower is better. The methods are summarized at the beginning of Sec. 5

Figure 3: Binary mixing macroscopic prediction by TAMPL. Particle colors indicate the two types at initial microstates. Curves show means, with shades indicating one standard deviation.

Figure 4: Microstates illustration and TAMPL predictions of polymer extension for the Fast, Medium, and Slow test cases.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Numerical evidence of the failure mode of joint training and the optimization behavior of TAMPL. Curves and bands are the mean and standard deviation of five independent runs.

Figure 6: Numerical evidence of Case II. Curves and bands are the mean and standard deviation of five independent runs.

Figure 7: Independent local test of Theorem 2 . The complete weighted residual norm contracts and latent covariance remains positive. Both algorithms start from the same full-matrix perturbations.

Figure 8: One SIRS microstate trajectory.

Figure 9: Comparison of different methods on SIRS prediction. The top row shows one reference trajectory given the initial microstate. The other rows compare the predicted and ground-truth macrostates. The reference simulations and all methods except VAMP use ensembles of $64$ independent trajectories initialized from the same microstate. All models were trained with seed $42$ .

Transition weight $\lambda_{\mathrm{tr}}$	Mean macrostate RMSE	MMD
$0.1$	$2.4854\pm 3.3309$	$0.3339\pm 0.2898$
$0.5$	$0.5779\pm 0.1032$	$0.1883\pm 0.0744$
$1.0$	$0.6254\pm 0.2400$	$0.2263\pm 0.1200$
$2.0$	$0.5323\pm 0.2071$	$0.2029\pm 0.0711$

Appendix

Table 3: Sensitivity of JLD to the transition-loss weight on SIRS, with

\lambda_{\mathrm{cur}}=1

. Values are the mean

\pm

sample standard deviation over training seeds

42

43

, and

44

Figure 10: One example of the microstate trajectory in binary mixing.

Figure 11: Comparison of methods for binary mixing prediction from two initial configurations. The top row shows snapshots from one reference trajectory for each configuration. Subsequent rows compare predicted and reference evolution of $S$ and $K$ . Curves show ensemble means, with shaded bands indicating one standard deviation. VAMP provides a deterministic prediction.

Figure 12: Comparison of methods for polymer extension prediction.

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