NEMSim: Learning Control-Conditioned Multi-Event Physical Dynamics via Executable Event-Mechanism Priors

Junsong Yu; Junjie Xie; Pengwei Liu; Dong Ni

cs.LGSep 25, 2026

NEMSim: Learning Control-Conditioned Multi-Event Physical Dynamics via Executable Event-Mechanism Priors

Authors: Junsong Yu, Junjie Xie, Pengwei Liu, Dong Ni

Organizations: College of Integrated Circuits, Zhejiang University · College of Control Science and Engineering, Zhejiang University

Abstract

High-fidelity simulation of control-conditioned multi-event physical systems is computationally expensive, especially across broad control spaces and long trajectories. In these systems, macroscopic evolution emerges from localized discrete events whose intensities and effects depend on process controls and evolving local states, while the available system knowledge is typically expressed as event-attribute descriptions. Purely data-driven surrogates must infer these event effects from limited trajectory coverage, which can hinder generalization to unseen control regimes. Physics-guided methods instead primarily build on equation-level constraints or differentiable solvers rather than discrete event-rule priors. We therefore propose NEMSim (Neural Event-Mechanism Simulator), which compiles predefined event-attribute descriptions into an executable transition structure linking control-dependent event intensities, prior-guided mechanism attribution, and state-dependent responses. To enable evaluation of control-conditioned multi-event dynamics with explicit system knowledge, we construct a 3D KMC-based benchmark pairing high-fidelity trajectories with explicit event rules, standardized splits, and evaluation protocols. Across three settings, NEMSim reduces Avg. RMSE by 58.9%-81.3% relative to the strongest baseline in each setting. It also remains best in the data-efficiency study with training-data fractions down to 10%. Mechanism analyses further show that these gains arise from executable rule integration rather than prior access or architecture alone.

Figures & tables

Figure 1: Overview of NEMSim and the representative control-conditioned multi-event system. The left panel uses plasma etching as an example to illustrate controlled multi-event system evolution. The right panel shows the NEMSim pipeline, which compiles $\mathbf{x}_{t}$ , $\mathbf{u}_{t}$ , and the event rule library into an executable event-mechanism transition structure to predict $\hat{\mathbf{x}}_{t+1}$ .

Figure 2: Detailed architecture of the EIC, MAC and MRC modules in NEMSim. Purple pentagons mark operations parameterized by event-rule attributes.

Method	Control-ID			OOD-Joint			Vis-50%			Params. $\downarrow$ (M)
Method	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$	Params. $\downarrow$ (M)
ConvLSTM	0.2730	0.4557	3.847	0.3657	0.6513	3.360	0.7117	0.8835	3.596	0.634
FNO	0.5263	0.7510	3.838	0.4318	0.5639	2.865	0.4747	0.5995	3.398	1.358
U-NO	0.4526	0.5403	3.849	0.5673	0.7861	3.567	0.4131	0.5069	3.515	2.432
CNO	0.1841	0.3027	2.005	0.2552	0.4078	2.282	0.1041	0.2275	1.307	1.734
DeepONet	0.0365	0.0842	0.987	0.1243	0.2120	1.122	0.1207	0.2004	1.202	1.720

Table 1: Predictive performance over 100-step autoregressive rollouts. Results for OOD-Ar + , OOD-Cl + , and Vis-60% are provided in the supplementary material (Sec. S5.1 and Table S11). Computational-efficiency profiling is provided in the supplementary material (Sec. S5.2 and Table S12).

Figure 3: Mean stepwise RMSE over 100-step autoregressive rollouts under OOD-Joint and Vis-50%.

Figure 4: Visualization of final-step rollout predictions under OOD-Joint and Vis-50%. Surface errors denote signed deviations from the ground-truth surface, and SDF errors are shown on representative slices. Additional qualitative results are provided in the supplementary material (Sec. S7).

Figure 5: Data efficiency under OOD-Joint across fractions of training trajectories. (a) Average RMSE over 100-step autoregressive rollouts. (b) Final-step Chamfer distance.

Variant	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$
NEMSim (Full)	0.0511	0.1141	0.851
Structure-only	0.0684	0.1525	0.944
Mechanism Shuffle	0.0683	0.1497	0.896
Event Drop	0.0892	0.2167	0.892
Rate/Yield Noise	0.0944	0.1812	1.117

Table 2: Event-rule ablation and perturbation results under OOD-Joint. Detailed configurations are provided in the supplementary material (Sec. S5.5).

Figure 6: Mechanism-aligned responses under counterfactual control-role perturbations. (a) Changes in the learned physical, chemical, and coupled responses. (b) Relationships between selected mechanism-response changes and material-removal changes.

Configuration	Implemented value	Interpretation
Native material domain	$52\times 52\times 125$	Multi-species voxel domain in $(x,y,z)$ ordering.
Nominal voxel spacing	$2\times 2\times 2~\mathrm{nm}$	Used for geometric scale and flux-to-particle conversion; level-set evolution uses lattice units.
Initial Si region	$z<80$	Bulk silicon substrate with nominal thickness $160~\mathrm{nm}$ .
Initial resist region	$80\leq z<120$	Resist layer with a centered cylindrical opening.
Opening radius	$10$ voxels	Nominal radius $20~\mathrm{nm}$ and diameter $40~\mathrm{nm}$ .
Incident region	$120\leq z<125$	Initially empty region above the resist surface.

Table S1: Physical and numerical configuration of the particle-based Monte Carlo reference simulator.

Index	Process variable	Physical role	Sampling transformation	Configured range
1	$\Gamma_{\mathrm{Ar}^{+}}$	Ar + incident flux	Logarithmic LHS	$10^{-4}$ – $7\times 10^{-3}~\mathrm{mol\,m^{-2}\,s^{-1}}$
2	$\Gamma_{\mathrm{Cl}^{+}}$	Cl + incident flux	Logarithmic LHS	$10^{-4}$ – $7\times 10^{-3}~\mathrm{mol\,m^{-2}\,s^{-1}}$
3	$\Gamma_{\mathrm{Cl}_{2}^{+}}$	Cl ${}_{2}^{+}$ incident flux	Logarithmic LHS	$10^{-6}$ – $2\times 10^{-4}~\mathrm{mol\,m^{-2}\,s^{-1}}$
4	$\Gamma_{\mathrm{Cl}}$	Neutral Cl incident flux	Logarithmic LHS	$10^{-3}$ – $2\times 10^{-2}~\mathrm{mol\,m^{-2}\,s^{-1}}$
5	$m_{\mathrm{Cl}}$	Neutral Cl angular concentration	Bounded integer LHS	$0$ – $20$
6	$m_{\mathrm{Cl}^{+}}$	Cl + angular concentration	Bounded integer LHS	$50$ – $3000$

Table S2: Definition and sampling ranges of the 14-dimensional process control space. Flux variables are sampled through logarithmic mappings, angular exponents through bounded integer mappings, and ion-energy statistics through linear mappings.

Property	Value
Number of trajectories	500
Unique control configurations	500
Saved states per trajectory	101
Transitions per trajectory	100
Total saved states	50,500
Total one-step transitions	50,000

Table S3: Summary statistics of the constructed benchmark before train, validation, and test splitting.

Setting	Train	Val.	Test	Unused	Trans./traj.	Train samples	Evaluation condition
Control-ID	300	100	100	0	100	30,000	Unseen control configurations within the benchmark control domain.
OOD-Ar +	280	70	150	0	100	28,000	Ar + incident flux outside its central training window.
OOD-Cl +	279	70	151	0	100	27,900	Cl + incident flux outside its central training window.
OOD-Joint	202	50	53	195	100	20,200	Ar + and Cl + fluxes simultaneously outside their central training windows.
Vis-50%	300	100	100	0	50	15,000	Only the first 50 transitions are visible during one-step training.
Vis-60%	300	100	100	0	60	18,000	Only the first 60 transitions are visible during one-step training.

Table S4: Trajectory-level splits for the complete 500-trajectory benchmark. “Trans./traj.” denotes the number of one-step transitions available to the training dataset. All final evaluations use complete 100-step autoregressive rollouts.

Released component	Representation	Description
State trajectories	$\boldsymbol{\Phi}\in\mathbb{R}^{101\times 2\times 127\times 54\times 54}$	Normalized mask and Si signed-distance fields for one complete trajectory, ordered as $(T,C,D,H,W)$ .
Raw controls	$\mathbf{u}^{\mathrm{raw}}\in\mathbb{R}^{14}$	Original process-control values in their physical or numerical units.
Transformed controls	$\widetilde{\mathbf{u}}\in\mathbb{R}^{14}$	Control vector after logarithmic transformation of the four flux dimensions and before range normalization.
Time coordinates	$\mathbf{t}\in\mathbb{R}^{101}$	Simulator macro-times associated with the saved states.
Case metadata	Case and control identifiers	Identifiers, parameter order, tensor dimensions, channel semantics, and representation metadata.
Event Rules Library	Machine-readable rule records	Fixed event attributes, kinetic parameters, dependency relations, and mechanism-family priors shared across all trajectories.

Table S5: Principal components of the released benchmark package.

Attribute group	Source fields	Executable role
Event identity	name , group , projectile , target , effect	Defines event identity, species semantics, target type, semantic mechanism support, and response-role flags.
Control binding	flux_key , angle_key , E_mean_key , E_std_key	Selects the event-specific supply, incidence, and energy channels gathered from $\mathbf{u}_{t}$ by EIC.
Probability	prob_model , prob_params	Selects a constant or energy-angle-dependent interaction probability and provides $p^{\mathrm{const}}$ or $E^{\mathrm{th}}$ , $E^{\mathrm{ref}}$ , $p^{0}$ , $n$ , and the angular-law type.
Yield	yield_model , yield_params	Selects a constant or Yamamura-style yield and provides $Y^{\mathrm{const}}$ or the yield threshold, maximum-yield angle, maximum yield, and normal-incidence yield.
Product source	products.species , products.coeff	Defines which intermediate species are generated and whether their amount is fixed or proportional to the event yield.
Product sink	dep_model , dep_params.species , dep_params.p_stick	Defines product-mediated deposition events and their required species and sticking coefficients.

Table S6: Event-rule attributes and their executable roles in NEMSim.

ID	Configured event	Surface role	$p_{e}^{\mathrm{const}}$	Effect	Pool contribution
E1	Cl_on_Si_to_SiCl	$\mathrm{Si}\rightarrow\mathrm{SiCl}$	$0.55$	React	$\mathrm{SiCl}$
E2	Cl_on_SiCl_to_SiCl2	$\mathrm{SiCl}\rightarrow\mathrm{SiCl}_{2}$	$0.30$	React	$\mathrm{SiCl}_{2}$
E3	Cl_on_SiCl2_to_SiCl3	$\mathrm{SiCl}_{2}\rightarrow\mathrm{SiCl}_{3}$	$0.20$	React	$\mathrm{SiCl}_{3}$
E4	Cl_on_SiCl2_to_SiCl_back	$\mathrm{SiCl}_{2}\rightarrow\mathrm{SiCl}$	$0.02$	React	$\mathrm{SiCl}$
E5	Cl_on_SiCl3_etch	Removal from $\mathrm{SiCl}_{3}$	$0.0001$	Etch	$\mathrm{SiCl}_{3}$
E6	Cl_on_SiCl3_to_SiCl2	$\mathrm{SiCl}_{3}\rightarrow\mathrm{SiCl}_{2}$	$0.08$	React	$\mathrm{SiCl}_{2}$

Table S7: Neutral-chlorine events in the benchmark Event Rules Library. All six channels use the neutral-Cl supply control and a constant unit yield.

ID	Projectile	Target / branch	Group	$\mathcal{Q}_{e}$	Ang.	$\mathcal{Y}_{e}$	Pool species	Coeff.
E7	$\mathrm{Cl}^{+}$	$\mathrm{Si}$ / P	physical	$(25,100,0.05)$	P	$(25,60,2.2,1.2)$	$\mathrm{SiCl}$	$Y$
E8	$\mathrm{Cl}^{+}$	$\mathrm{SiCl}$ / P	both	$(35,100,0.10)$	P	$(35,60,2.8,1.5)$	$\mathrm{SiCl}_{2}$	$Y$
E9	$\mathrm{Cl}^{+}$	$\mathrm{SiCl}$ / C	both	$(10,100,0.20)$	C	$1$	$\mathrm{SiCl}_{2}$	$1$
E10	$\mathrm{Cl}^{+}$	$\mathrm{SiCl}_{2}$ / C	both	$(10,100,0.50)$	C	$1$	$\mathrm{SiCl}_{2}$	$1$
E11	$\mathrm{Cl}^{+}$	$\mathrm{SiCl}_{3}$ / C	both	$(10,100,0.50)$	C	$1$	$\mathrm{SiCl}_{3}$	$1$
E12	$\mathrm{Cl}^{+}$	Resist / P	physical	$(15,150,0.01)$	P	$(15,60,1.2,1.0)$	$\mathrm{SiCl}$	$Y$

Table S8: Ion-induced primary events in the benchmark Event Rules Library. For all channels,

n_{e}=0.5

. The YAML label both identifies events assigned to the coupled static mechanism prior.

Pool species	Target classes	Channels	$p^{\mathrm{stick}}$
$\mathrm{SiCl}$	All five	E23–E27	$0.02$
$\mathrm{SiCl}_{2}$	All five	E28–E32	$0.02$
$\mathrm{SiCl}_{3}$	All five	E33–E37	$0.02$

Table S9: Product-mediated deposition events. “All five” denotes

\{\mathrm{Si},\mathrm{SiCl},\mathrm{SiCl}_{2},\mathrm{SiCl}_{3},\mathrm{Resist}\}

Mechanism	Functional interpretation	Compiled rule evidence	Representative event roles
Physical	Direct energy or momentum transfer from an energetic projectile to the surface.	Ion-driven event; physical angular type P ; Yamamura-style yield; sputtering or resist-erosion role.	Direct sputtering of Si; physical removal from a surface; energetic resist erosion.
Chemical	Surface transformation governed primarily by reactant availability and surface composition.	Neutral-driven reaction; chemical angular type C ; chlorination, abstraction, spontaneous reaction, or deposition role.	Si chlorination; Cl abstraction; spontaneous removal; SiCl x attachment.
Coupled	Response requiring interaction between energetic activation, surface chemistry, or product-mediated event dependencies.	Rule group both ; ion interaction with a chlorinated target; product-source or product-sink relation.	Ion-assisted removal of SiCl x ; chemically enhanced sputtering; generation and redeposition of products.

Table S10: Mechanism-family interpretation and the event-rule attributes used by the compiler. The listed attributes provide structural evidence rather than event-level supervision.

Setting	Method	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$
OOD-Ar +	ConvLSTM	0.7710	0.8722	3.575
	FNO	0.6109	0.7843	3.398
	U-NO	0.5835	0.7687	3.272
	CNO	0.2535	0.4253	3.241
	DeepONet	0.1055	0.0833	0.755
	NEMSim (ours)	0.0218	0.0509	0.586

Table S11: Additional 100-step autoregressive rollout results under single-axis Control-OOD and Vis-60% temporal extrapolation. The best result in each setting is shown in bold.

Method	GFLOPs/step	Mean (ms) $\downarrow$	P95 (ms) $\downarrow$	Throughput (s -1 ) $\uparrow$	Peak mem. (MiB) $\downarrow$
NEMSim	41.688	11.439	11.442	87.42	330.380
ConvLSTM	331.388	4.541	4.970	220.19	632.422
CNO	159.410	10.960	11.218	91.24	336.233
U-NO	1.061	3.579	3.620	279.43	183.034
FNO	4.473	2.149	2.154	465.41	214.918
DeepONet	131.081	3.091	3.104	323.51	590.570

Table S12: One-step inference efficiency at the standardized

2\times 64\times 64\times 64

input resolution. Latency and throughput are measured with a batch size of one. Mean and P95 report latency in milliseconds. Throughput is measured in samples per second. Peak memory denotes the maximum GPU memory allocated by PyTorch during one synchronized forward pass.

Data	Method	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$
$10\%$	DeepONet	0.1955	0.2853	2.609
	DeepONet+Prior	0.1975	0.2660	2.902
	NEMSim	0.0967	0.2097	1.167
$20\%$	DeepONet	0.1906	0.2239	2.924
	DeepONet+Prior	0.1755	0.2331	1.978
	NEMSim	0.0902	0.1936	1.095

Table S13: Data-efficiency results under OOD-Joint. Training-data fractions denote percentages of the original training trajectories.

Setting	Variant	Avg. RMSE $\downarrow$	Final RMSE $\downarrow$	CD $\downarrow$
Control-ID	Full	0.0134	0.0316	0.519
	w/o EIC	0.0612	0.1282	0.881
	w/o MAC	0.0611	0.1234	0.828
	w/o MRC	0.0600	0.1258	0.827
OOD-Joint	Full	0.0511	0.1141	0.851
	w/o EIC	0.1229	0.2549	1.207

Table S14: Ablation results under Control-ID, OOD-Joint, and Vis-50% settings.

Method	3D adaptation	Control injection	Core configuration	Params (M)
ConvLSTM	Four stacked ConvLSTM3D layers followed by a 3D convolutional readout.	The projected control vector is spatially broadcast and concatenated with every input state.	Four layers with hidden widths $(28,28,28,28)$ .	0.634
FNO	3D Fourier layers operate along all three spatial dimensions, with a pointwise residual path in each operator block.	A projected control field is broadcast and concatenated with the input state before lifting.	Width $14$ , four Fourier blocks, and $6\times 6\times 6$ retained modes.	1.358
U-NO	The seven-layer U-shaped neural-operator topology is extended to 3D using dense 3D Fourier operators and multiresolution skip connections.	A projected control field is broadcast and concatenated with the state and spatial coordinates before lifting.	Widths $(8,16,32)$ and $4\times 4\times 4$ Fourier modes.	2.432
CNO	The Lift–Encoder–ResNet–ED Link–Decoder–Project topology is extended to 3D. The implementation applies alias-controlled activations only at resolution-changing operations.	A projected control field is broadcast and concatenated with the state and spatial coordinates before the lifting block.	Widths $(22,44,88)$ , two residual blocks per stage, and an oversampling factor of $1.5$ .	1.734
DeepONet	A 3D convolutional branch encoder is combined with a coordinate-based trunk network through a branch–trunk basis contraction.	The control vector is projected into a spatial feature map and concatenated with the state inside the branch encoder.	Branch widths $(32,64,128)$ , basis dimension $192$ , and a three-layer trunk with width $160$ .	1.720

Table S15: Implementation details of the data-driven baselines. Parameter counts include all trainable parameters. Complex-valued Fourier coefficients are counted as two real-valued parameters.

Figure S1: Additional qualitative comparison under OOD-Joint. Rows correspond to autoregressive rollout steps $10$ , $25$ , $50$ , $75$ , and $100$ . The left block compares predicted surfaces colored by signed surface deviation, while the right block shows representative SDF errors and zero-level contours. NEMSim maintains closer agreement with the ground-truth morphology and contour throughout the rollout, whereas the baselines exhibit increasing bottom, sidewall, and global shape deviations.

Figure S2: Additional qualitative comparison under Vis-50%. Rows correspond to autoregressive rollout steps $10$ , $25$ , $50$ , $75$ , and $100$ . Steps $75$ and $100$ lie beyond the temporal range exposed during training. The left block compares signed surface deviations, and the right block compares SDF errors and zero-level contours. NEMSim better preserves the evolving trench geometry after entering the temporally extrapolative stage.

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