Adaptive Interaction Graphs for Particle Simulation

Aiden Zhou

cs.LGSep 25, 2026

Adaptive Interaction Graphs for Particle Simulation

Authors: Aiden Zhou

Organizations: Yale University, New Haven, Connecticut, USA.

Abstract

Learned particle simulators based on graph neural networks achieve strong one-step accuracy, but errors compound over long horizons. An underexplored variable is the interaction graph: existing methods fix its topology via k-nearest neighbors or a static radius rule, regardless of local model confidence. We propose making this graph adaptive: a per-particle variance head, trained jointly with the acceleration head under a heteroscedastic Gaussian NLL loss, drives a trajectory in which high-uncertainty particles receive an expanded neighborhood. This is done at little extra inference cost by using the previous step's uncertainty estimate. A key discovery is that the variance head learns a meaningful notion of uncertainty: high-variance particles concentrate near complex regions, such as splash zones or free surfaces. When this signal drives graph topology, the resulting AdaptGNS simulator achieves a strict Pareto improvement on WaterDrop and a modest gain on Sand. Given the model's stronger performance on WaterDrop, we hypothesize that adaptive graphs are most useful when complexity is concentrated in space. Our code can be found at https://github.com/aidenzhou8/AdaptGNS.

Figures & tables

Figure 1: Interaction graph snapshots at Step 100 for WaterDrop (left) and Sand (right).

Dataset	Train	Val.	Test	Steps
Sand	1,000	100	30	320
WaterDrop	1,000	30	30	1,000

Table 1: Data splits and rollout lengths.

Dataset	Trajs	Nodes	Edges	Density	Avg Degree	Avg Clustering
WaterDrop	30	$567\pm 200$	$1550\pm 583$	$1.08\times 10^{-2}$	$5.42\pm 0.22$	$0.494\pm 0.036$
Sand	30	$1270\pm 503$	$4324\pm 1864$	$7.35\times 10^{-3}$	$6.76\pm 1.01$	$0.500\pm 0.034$

Table 2: Graph statistics at Step 100, averaged over 30 test trajectories.

Figure 2: Per-particle uncertainty $\sigma_{i}$ at steps 1, 10, and 50 of a Sand trajectory. High- $\sigma$ particles are concentrated in impact zones, free surfaces, and regions of rapid velocity change.

	Sand			WaterDrop
Bin	mean SE	$2\hat{\sigma}^{2}$	$\|\Delta\|$	mean SE	$2\hat{\sigma}^{2}$	$\|\Delta\|$
0	0.023	0.009	0.014	0.543	0.003	0.540
1	0.025	0.014	0.011	0.540	0.004	0.537
2	0.024	0.018	0.006	0.131	0.009	0.121
3	0.026	0.023	0.003	0.081	0.015	0.066
4	0.033	0.029	0.004	0.208	0.022	0.186

Table 3: Per-decile calibration on Sand and WaterDrop .

Figure 3: Per-step dynamics (edge count and MSE) on Sand and WaterDrop .

Physics-Based Simulation

Physics-based simulation models physical systems with equations and numerical solvers, increasingly augmented by learned models for speed and control.

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Abstract

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