$S^3$: A Smooth Simulation Surrogate for Optimizing Discrete Abstractions of Dynamical Systems

Jordan Peper; James Mathias Gast; Vignesh Nanduri; Tanmayee Maram; Ethan Howes; Ivan Ruchkin

eess.SYAug 16, 2026

$S^3$ : A Smooth Simulation Surrogate for Optimizing Discrete Abstractions of Dynamical Systems

Authors: Jordan Peper, James Mathias Gast, Vignesh Nanduri, Tanmayee Maram, Ethan Howes, Ivan Ruchkin

Organizations: University of Florida

Abstract

Intelligent systems are increasingly deployed in safety-critical settings with black-box controllers, including neural networks. The properties and behaviors of these end-to-end systems can be studied with abstraction-based methods that replace them with simpler finite models. Constructing such abstractions requires balancing the soundness of over-approximating the dynamical system against conservatism, which manifests as spurious or excessive nondeterministic behaviors. Bi-simulation theory provides principled metrics for characterizing these relationships, but does not prescribe how to construct sound abstractions with minimal conservatism. We fill this gap with a smooth simulation surrogate ( $S^3$ ) --- a differentiable objective that approximates the reverse simulation metric used to quantify conservatism. Combined with Taylor model-based reachability, $S^3$ enables gradient-based optimization of abstraction parameters while preserving soundness by construction. We evaluate this optimization pipeline on three case studies. Our results show that $S^3$ is strongly correlated with the reverse simulation metric, is computationally faster, and serves as an effective objective for reducing abstraction conservatism.

Figures & tables

Figure 1: The quantization process forming variable-width cells from unconstrained parameters $\omega_{i}$ in $\theta$ .

		(Max) $\sigma^{\leftarrow}_{H}$		(Mean) $\overline{\delta}_{H}$		(Median) $\delta^{0.50}_{H}$		Time per sample (s)
	$H$	$r_{p}\uparrow$	$r_{s}\uparrow$	$r_{p}\uparrow$	$r_{s}\uparrow$	$r_{p}\uparrow$	$r_{s}\uparrow$	(Real) $T\downarrow$	(Proxy) $\hat{T}\downarrow$
Spiral	1	$0.88^{+0.03}_{-0.06}$	$0.85^{+0.06}_{-0.08}$	$0.45^{+0.12}_{-0.15}$	$0.42^{+0.14}_{-0.16}$	—	—	$2.95^{+0.05}_{-0.04}$	$0.01^{+0.03}_{-0.01}$
	2	$0.78^{+0.10}_{-0.10}$	$0.70^{+0.10}_{-0.14}$	$0.69^{+0.10}_{-0.14}$	$0.67^{+0.10}_{-0.14}$	$0.47^{+0.13}_{-0.19}$	$0.47^{+0.15}_{-0.19}$	$3.08^{+0.04}_{-0.04}$	$0.01^{+0.03}_{-0.01}$
	3	$0.73^{+0.08}_{-0.11}$	$0.75^{+0.08}_{-0.11}$	$0.66^{+0.09}_{-0.14}$	$0.66^{+0.11}_{-0.15}$	$0.57^{+0.12}_{-0.17}$	$0.56^{+0.14}_{-0.18}$	$3.61^{+0.06}_{-0.05}$	$0.01^{+0.03}_{-0.01}$
	4	$0.62^{+0.13}_{-0.15}$	$0.65^{+0.10}_{-0.14}$	$0.64^{+0.10}_{-0.15}$	$0.65^{+0.11}_{-0.15}$	$0.56^{+0.12}_{-0.16}$	$0.59^{+0.12}_{-0.17}$	$4.11^{+0.06}_{-0.06}$	$0.01^{+0.03}_{-0.01}$
	5	$0.52^{+0.17}_{-0.17}$	$0.55^{+0.13}_{-0.17}$	$0.63^{+0.10}_{-0.15}$	$0.65^{+0.11}_{-0.15}$	$0.57^{+0.13}_{-0.19}$	$0.60^{+0.12}_{-0.16}$	$4.30^{+0.06}_{-0.06}$	$0.01^{+0.03}_{-0.01}$

Table 1: Correlation coefficients between

S^{3}

and simulation metrics, and their evaluation time on randomly sampled parameters.

			Sim. Metrics			Task	Time (s)
	$m_{i}$		$\sigma^{\leftarrow}_{H}\downarrow$	$\overline{\delta}_{H}\downarrow$	$\delta_{H}^{0.50}\downarrow$	Recall $\uparrow$	$T_{b}\downarrow$	$T_{v}\downarrow$	$T_{t}\downarrow$
Spiral	70	(a)	2.020	0.373	0.388	0.916	8.520	0.032	8.552
		(b)	2.064	0.372	0.429	0.872	0.140	0.030	0.170
		(c)	2.064	0.372	0.429	0.872	0.130	0.020	0.150
	100	(a)	2.120	0.264	0.281	0.939	24.14	0.045	24.19
		(b)	2.130	0.268	0.300	0.928	0.250	0.090	0.340

Table 2: Model performance across three methods: (a) CEGAR, (b) Weber ( 2017 ) , (c)

S^{3}

+ SGD (ours).

Figure 1: Displacement field of the spiral system. Blue circle is the goal set.

Figure 2: Displacement field of the unicycle system for the $x_{3}=0$ (heading angle) slice. Red circle is the obstacle, and blue circle is the goal set.

Figure 3: Displacement field of the Mountain Car under the pretrained DDPG. Blue region is the goal set ( $x_{1}\geq 0.45$ ).

		$\sigma^{\leftarrow}_{H}$		$\overline{\delta}_{H}$		$\delta^{0.50}_{H}$		Time per sample (s)
	$H$	$r_{p}$	$r_{s}$	$r_{p}$	$r_{s}$	$r_{p}$	$r_{s}$	$T$	$\hat{T}$
Spiral	1	$0.76^{+0.08}_{-0.10}$	$0.72^{+0.09}_{-0.12}$	$0.50^{+0.12}_{-0.14}$	$0.48^{+0.14}_{-0.16}$	$-$	$-$	$5.07^{+0.19}_{-0.21}$	$0.01^{+0.03}_{-0.01}$
	2	$0.72^{+0.10}_{-0.12}$	$0.65^{+0.11}_{-0.15}$	$0.81^{+0.06}_{-0.09}$	$0.79^{+0.07}_{-0.10}$	$0.70^{+0.08}_{-0.12}$	$0.68^{+0.10}_{-0.13}$	$5.25^{+0.18}_{-0.21}$	$0.01^{+0.04}_{-0.01}$
	3	$0.70^{+0.10}_{-0.14}$	$0.67^{+0.11}_{-0.15}$	$0.78^{+0.07}_{-0.10}$	$0.76^{+0.08}_{-0.12}$	$0.72^{+0.08}_{-0.12}$	$0.70^{+0.10}_{-0.13}$	$6.05^{+0.21}_{-0.24}$	$0.01^{+0.04}_{-0.01}$
	4	$0.60^{+0.13}_{-0.18}$	$0.62^{+0.12}_{-0.17}$	$0.74^{+0.07}_{-0.10}$	$0.74^{+0.08}_{-0.12}$	$0.70^{+0.09}_{-0.12}$	$0.71^{+0.09}_{-0.13}$	$6.85^{+0.23}_{-0.27}$	$0.01^{+0.04}_{-0.01}$
	5	$0.50^{+0.16}_{-0.18}$	$0.52^{+0.14}_{-0.18}$	$0.73^{+0.07}_{-0.10}$	$0.73^{+0.09}_{-0.12}$	$0.70^{+0.09}_{-0.15}$	$0.72^{+0.09}_{-0.13}$	$7.23^{+0.25}_{-0.28}$	$0.01^{+0.03}_{-0.01}$

Table 3: Correlation coefficients between

S^{3}

and simulation metrics, and their evaluation time on randomly sampled parameters. LSE temperatures are set to

\tau_{1}=\tau_{2}=0.5

		$\sigma^{\leftarrow}_{H}$		$\overline{\delta}_{H}$		$\delta^{0.50}_{H}$		Time per sample (s)
	$H$	$r_{p}$	$r_{s}$	$r_{p}$	$r_{s}$	$r_{p}$	$r_{s}$	$T$	$\hat{T}$
Spiral	1	$0.70^{+0.09}_{-0.12}$	$0.67^{+0.10}_{-0.13}$	$0.48^{+0.12}_{-0.15}$	$0.47^{+0.14}_{-0.16}$	$-$	$-$	$2.91^{+0.05}_{-0.04}$	$0.01^{+0.04}_{-0.01}$
	2	$0.66^{+0.10}_{-0.13}$	$0.61^{+0.13}_{-0.17}$	$0.81^{+0.06}_{-0.09}$	$0.78^{+0.07}_{-0.10}$	$0.73^{+0.08}_{-0.11}$	$0.71^{+0.09}_{-0.12}$	$3.06^{+0.04}_{-0.04}$	$0.01^{+0.03}_{-0.01}$
	3	$0.65^{+0.11}_{-0.14}$	$0.63^{+0.12}_{-0.16}$	$0.78^{+0.07}_{-0.10}$	$0.76^{+0.08}_{-0.11}$	$0.73^{+0.08}_{-0.11}$	$0.72^{+0.09}_{-0.12}$	$3.54^{+0.06}_{-0.05}$	$0.01^{+0.04}_{-0.01}$
	4	$0.56^{+0.13}_{-0.19}$	$0.59^{+0.13}_{-0.18}$	$0.74^{+0.07}_{-0.10}$	$0.73^{+0.08}_{-0.12}$	$0.72^{+0.08}_{-0.11}$	$0.72^{+0.09}_{-0.12}$	$4.04^{+0.12}_{-0.06}$	$0.01^{+0.03}_{-0.01}$
	5	$0.47^{+0.15}_{-0.19}$	$0.50^{+0.14}_{-0.19}$	$0.73^{+0.07}_{-0.10}$	$0.72^{+0.09}_{-0.12}$	$0.72^{+0.08}_{-0.12}$	$0.72^{+0.09}_{-0.13}$	$4.23^{+0.10}_{-0.07}$	$0.01^{+0.03}_{-0.01}$

Table 4: Correlation coefficients between

S^{3}

and simulation metrics, and their evaluation time on randomly sampled parameters. LSE temperatures are set to

\tau_{1}=\tau_{2}=1.0

Figure 4: All orthogonal projections of the final grid after optimization of Unicycle ( $m_{i}=100$ , $H=5,\tau_{1}=\tau_{2}=0.1$ ).

Figure 5: Final grid after optimization of mountain car ( $m_{i}=100$ , $H=1,\tau=0.1$ ).

			Sim. Metrics						Task
	$m_{i}$	$(H,\tau)$	$\sigma^{\leftarrow}_{H}\downarrow$	$\Delta\sigma^{\leftarrow}_{H}\downarrow$	$\overline{\delta}_{H}\downarrow$	$\Delta\overline{\delta}_{H}\downarrow$	$\delta_{H}^{0.5}\downarrow$	$\Delta\delta_{H}^{0.5}\downarrow$	Recall $\uparrow$	$\Delta\text{Recall}\uparrow$
Spiral	70	$(1,0.1)$	$0.14$	$\scalebox{0.6}[1]{$ - $}0.021$	$0.027$	$\scalebox{0.6}[1]{$ - $}0.00067$	$0.0$	$0.0$	$0.87$	$0.011$
		$(1,1.0)$	$0.14$	$\scalebox{0.6}[1]{$ - $}0.020$	$0.027$	$\scalebox{0.6}[1]{$ - $}0.00067$	$0.0$	$0.0$	$0.87$	$0.011$
		$(3,0.1)$	$2.1$	$\scalebox{0.6}[1]{$ - $}0.024$	$0.37$	$\scalebox{0.6}[1]{$ - $}0.0045$	$0.43$	$0.013$	$0.87$	$0.011$
		$(3,1.0)$	$2.1$	$\scalebox{0.6}[1]{$ - $}0.023$	$0.37$	$\scalebox{0.6}[1]{$ - $}0.0043$	$0.43$	$0.013$	$0.87$	$0.011$
		$(5,0.1)$	$3.9$	$\scalebox{0.6}[1]{$ - $}0.029$	$0.67$	$\scalebox{0.6}[1]{$ - $}0.0065$	$0.71$	$0.012$	$0.87$	$0.011$

Table 5:

S^{3}

+ SGD optimization results for multiple horizons

H

and temperatures

\tau_{1}=\tau_{2}=\tau

Explore similar work

Abstract Sim2Real through Approximate Information States
Apr 16, 2026Yunfu Deng, Yuhao Li, Josiah P. HannaSim-to-Real Transfer State Abstraction
Department of Computer Sciences, University of Wisconsin–Madison, Madison, WI 53706 USA · Manning College of Information and Computer Sciences, University of Massachusetts Amherst, Amherst, MA 01003 USA
Towards Infinitely Long Neural Simulations: Self-Refining Neural Surrogate Models for Dynamical Systems
Mar 18, 2026Qi Liu, Laure Zanna, Joan BrunaNeural Surrogate Modeling Autoregressive Modeling
Courant Institute School of Mathematics, Computing, and Data Science, New York University, USA.
Joint Surrogate Learning of Objectives, Constraints, and Sensitivities for Efficient Multi-objective Optimization of Neural Dynamical Systems
Mar 22, 2026Frithjof Gressmann, Ivan Georgiev Raikov, Seung Hyun Kim +3Neural Surrogate Modeling Surrogate-Assisted Optimization
Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign, Urbana, IL · The Grainer College of Engineering, University of Illinois Urbana-Champaign, Urbana, IL · Department of Neurosurgery, Stanford University, Stanford, CA +3

Apr 16, 2026J/K move · Enter open · S save

Abstract Sim2Real through Approximate Information States

Yunfu Deng, Yuhao Li, Josiah P. Hanna

Department of Computer Sciences, University of Wisconsin–Madison, Madison, WI 53706 USA · Manning College of Information and Computer Sciences, University of Massachusetts Amherst, Amherst, MA 01003 USA

Sim-to-Real Transfer State Abstraction Robotic RL

Open paper AskPDF arXiv

In recent years, reinforcement learning (RL) has shown remarkable success in robotics when a fast and accurate simulator is available for a given task. When using RL and simulation, more simulator realism is generally beneficial but becomes harder to obtain as robots are deployed in increasingly complex and widescale domains. In such settings, simulators will likely fail to model all relevant details of a given target task and this observation motivates the study of sim2real with simulators that leave out key task details. In this paper, we formalize and study the abstract sim2real problem: given an abstract simulator that models a target task at a coarse level of abstraction, how can we train a policy with RL in the abstract simulator and successfully transfer it to the real-world? Our first contribution is to formalize this problem using the language of state abstraction from the RL literature. This framing shows that an abstract simulator can be grounded to match the target task if the grounded abstract dynamics take the history of states into account. Based on the formalism, we then introduce a method that uses real-world task data to correct the dynamics of the abstract simulator. We then show that this method enables successful policy transfer both in sim2sim and sim2real evaluation.

Abstract

Explore similar work

Abstract Sim2Real through Approximate Information States

Towards Infinitely Long Neural Simulations: Self-Refining Neural Surrogate Models for Dynamical Systems

Joint Surrogate Learning of Objectives, Constraints, and Sensitivities for Efficient Multi-objective Optimization of Neural Dynamical Systems