Benchmarking Optimizers to Solve Inverse Problems with Differentiable Physics Simulators

Xiang Chen; Huanhuan Xia

cs.LGSep 12, 2026

Benchmarking Optimizers to Solve Inverse Problems with Differentiable Physics Simulators

Authors: Xiang Chen, Huanhuan Xia

Organizations: Zhongguancun Academy, Beijing, China · Zhongguancun Institute of Artificial Intelligence, Beijing, China

Abstract

Solving inverse problems with differentiable physics simulators holds the potential to revolutionize scientific discovery and engineering design, as it enjoys both the strict physical correctness from rigorous numerical physics simulators, and the high efficiency and effectiveness from automatic differentiation and gradient-based optimization. However, currently, this paradigm faces performance issues in optimization. In this work, we target benchmarking the performance of different optimizers to solve various inverse problems. We construct 12 differentiable physics simulators spanning physics domains including discrete mechanics, continuous mechanics, atomistic simulations, rendering, and semi-empirical physics models. Based on these simulators, we design corresponding inverse problems that can be categorized into parameter identification, inverse design, and optimal control. Finally, we conduct extensive experiments to compare the performance of different optimizers, including regular first-order methods, approximate second-order methods, as well as global optimizers, on these inverse problems, and analyze the results to provide insights on how to choose and design optimizers for differentiable programming. We hope such benchmarks can inspire the development of more effective optimizers, and further promote the applications of differentiable programming in various scientific and engineering domains.

Figures & tables

Category	Model	Input	Output
Discrete Mechanics	Projectile Motion	Earth’s gravity initial state	horizontal range
	Flight Dynamics	vehicle params Earth params initial state control policy	flight trajectory
	Mass-Spring Systems	system params initial state control policy	movement trajectory
	Rigid-Body Dynamics	system params initial state control policy	movement trajectory
Continuous Mechanics	Elastic Mechanics	material density boundary conditions physical params	compliance displacement
Continuous Mechanics	Fluid Mechanics	car shape physical params	drag

Table 1: List of the implemented differentiable physics models.

Panel of Figure 1 — Figure 1: Gallery of the implemented differentiable physics simulators.

Category	Problem	Base Simulator	Dimension of Design Variables
Parameter Identification	XC Functional Fitting	DFT	2
	Tire Params Fitting	Tire Magic Formula	8
	Battery Params Fitting	Battery Thevenin Model	9
	Tight Binding Params Fitting	Tight Binding	32
	2D Reconstruction	Gaussian Splatting	224
Inverse Design	Range Matching	Projectile Motion	1

Table 2: List of the benchmarked inverse problems.

Category	Optimizer
First-order Local	SGD
	RMSprop
	Adam
	AdamW
Approximate Second-order Local	L-BFGS
	K-FAC

Table 3: List of the benchmarked optimizers.

Optimizer	DFT	Tire	Battery	Tight Binding	Gaussian Splatting
SGD	$1.9\text{e}^{-14}\pm 2.7\text{e}^{-14}$	$3.8\text{e}^{-4}\pm 1.9\text{e}^{-5}$	$2.0\text{e}^{-5}\pm 1.4\text{e}^{-5}$	$3.2\text{e}^{-5}\pm 2.1\text{e}^{-6}$	$2.2\text{e}^{-2}\pm 4.7\text{e}^{-4}$
RMSprop	$0.0\pm 0.0$	$2.6\text{e}^{-4}\pm 3.7\text{e}^{-5}$	$2.5\text{e}^{-4}\pm 2.2\text{e}^{-4}$	$5.6\text{e}^{-5}\pm 1.2\text{e}^{-5}$	$1.6\text{e}^{-2}\pm 1.6\text{e}^{-4}$
Adam	$1.9\text{e}^{-14}\pm 2.7\text{e}^{-14}$	$2.6\text{e}^{-4}\pm 1.7\text{e}^{-5}$	$5.1\text{e}^{-5}\pm 3.3\text{e}^{-5}$	$7.2\text{e}^{-7}\pm 2.8\text{e}^{-7}$	$1.4\text{e}^{-2}\pm 8.8\text{e}^{-5}$
AdamW	$3.5\text{e}^{-12}\pm 3.0\text{e}^{-12}$	$3.4\text{e}^{-4}\pm 1.5\text{e}^{-5}$	$3.3\text{e}^{-4}\pm 3.6\text{e}^{-4}$	$8.5\text{e}^{-7}\pm 4.9\text{e}^{-7}$	$1.4\text{e}^{-2}\pm 1.6\text{e}^{-4}$
LBFGS	$8.8\text{e}^{-11}\pm 6.2\text{e}^{-11}$	$2.5\text{e}^{-7}\pm 4.0\text{e}^{-8}$	$3.3\text{e}^{-6}\pm 5.2\text{e}^{-7}$	$1.1\text{e}^{-9}\pm 5.2\text{e}^{-10}$	$1.0\text{e}^{-2}\pm 1.9\text{e}^{-4}$
KFAC	$5.1\text{e}^{-10}\pm 1.3\text{e}^{-10}$	$2.4\text{e}^{-4}\pm 1.9\text{e}^{-5}$	$4.9\text{e}^{-5}\pm 5.8\text{e}^{-5}$	$3.5\text{e}^{-7}\pm 6.0\text{e}^{-8}$	$2.2\text{e}^{-2}\pm 1.1\text{e}^{-3}$

Table 4: The performance of optimizers on the benchmarked parameter identification problems.

Optimizer	Projectile	Fluid	Atmospheric	Elastic
SGD	$1.0\text{e}^{-9}\pm 1.3\text{e}^{-9}$	$6.2\text{e}^{-2}\pm 4.6\text{e}^{-7}$	$-7.5\text{e}^{-1}\pm 2.0\text{e}^{-3}$	$1.5\text{e}^{0}\pm 4.8\text{e}^{-3}$
RMSprop	$1.5\text{e}^{-9}\pm 1.9\text{e}^{-9}$	$4.2\text{e}^{-2}\pm 1.7\text{e}^{-7}$	$-7.5\text{e}^{-1}\pm 9.6\text{e}^{-5}$	$1.6\text{e}^{0}\pm 5.5\text{e}^{-3}$
Adam	$2.0\text{e}^{-10}\pm 1.5\text{e}^{-10}$	$4.0\text{e}^{-2}\pm 1.3\text{e}^{-7}$	$-7.5\text{e}^{-1}\pm 6.2\text{e}^{-4}$	$1.5\text{e}^{0}\pm 5.2\text{e}^{-3}$
AdamW	$2.9\text{e}^{-11}\pm 2.1\text{e}^{-11}$	$4.2\text{e}^{-2}\pm 2.3\text{e}^{-7}$	$-7.4\text{e}^{-1}\pm 1.9\text{e}^{-3}$	$1.5\text{e}^{0}\pm 6.0\text{e}^{-3}$
LBFGS	$2.4\text{e}^{-10}\pm 1.7\text{e}^{-10}$	$4.0\text{e}^{-2}\pm 2.1\text{e}^{-8}$	$-7.5\text{e}^{-1}\pm 4.9\text{e}^{-8}$	$1.5\text{e}^{0}\pm 4.5\text{e}^{-3}$
KFAC	$5.7\text{e}^{-10}\pm 1.9\text{e}^{-10}$	$4.0\text{e}^{-2}\pm 9.8\text{e}^{-9}$	$-7.5\text{e}^{-1}\pm 2.1\text{e}^{-5}$	$1.6\text{e}^{0}\pm 1.2\text{e}^{-2}$

Table 5: The performance of optimizers on the benchmarked inverse design problems.

Optimizer	Flight Dynamics	Rigid Body	Mass Spring
SGD	$-1.3\text{e}^{0}\pm 1.2\text{e}^{-6}$	$1.9\text{e}^{-1}\pm 1.7\text{e}^{-5}$	$1.9\text{e}^{-1}\pm 1.1\text{e}^{-2}$
RMSprop	$-1.3\text{e}^{0}\pm 6.7\text{e}^{-7}$	$1.9\text{e}^{-1}\pm 1.3\text{e}^{-5}$	$5.9\text{e}^{-2}\pm 9.6\text{e}^{-3}$
Adam	$-1.3\text{e}^{0}\pm 9.7\text{e}^{-8}$	$1.9\text{e}^{-1}\pm 2.7\text{e}^{-5}$	$1.3\text{e}^{-1}\pm 1.3\text{e}^{-2}$
AdamW	$-1.3\text{e}^{0}\pm 3.5\text{e}^{-7}$	$1.9\text{e}^{-1}\pm 1.5\text{e}^{-5}$	$1.4\text{e}^{-1}\pm 1.9\text{e}^{-2}$
LBFGS	$-1.3\text{e}^{0}\pm 5.0\text{e}^{-6}$	$1.9\text{e}^{-1}\pm 1.6\text{e}^{-3}$	$2.0\text{e}^{-1}\pm 3.0\text{e}^{-2}$
KFAC	$-1.3\text{e}^{0}\pm 5.6\text{e}^{-8}$	$1.9\text{e}^{-1}\pm 7.8\text{e}^{-6}$	$1.9\text{e}^{-1}\pm 1.0\text{e}^{-2}$

Table 6: The performance of optimizers on the benchmarked optimal control problems.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Name	Searched / fixed configurations
SGD	search lr , momentum (8 configs)
RMSprop	search lr , momentum (8 configs)
Adam	search lr , beta1 ; fixed beta2=0.999 (8 configs)
AdamW	search lr , beta1 ; fixed beta2=0.999 (8 configs)
LBFGS	search lr ; fixed max_iter=10 , line_search_fn=strong_wolfe (4 configs)
KFAC	search lr , damping ; fixed momentum=0.9 , beta=0.95 , precondition_frequency=10 (12 configs)

Appendix

Table 7: Optimizer hyper-parameter search space for the benchmark.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$1.1\times 10^{0}\pm 3.1\times 10^{-2}$	$5.1\times 10^{1}\pm 0.0$
RMSprop	$1.1\times 10^{0}\pm 8.3\times 10^{-3}$	$5.1\times 10^{1}\pm 0.0$
Adam	$9.6\times 10^{-1}\pm 1.1\times 10^{-2}$	$5.1\times 10^{1}\pm 0.0$
AdamW	$9.6\times 10^{-1}\pm 1.9\times 10^{-2}$	$5.1\times 10^{1}\pm 0.0$
LBFGS	$6.4\times 10^{-1}\pm 1.6\times 10^{-2}$	$5.1\times 10^{1}\pm 1.8\times 10^{-3}$
KFAC	$1.1\times 10^{0}\pm 1.2\times 10^{-2}$	$5.1\times 10^{1}\pm 0.0$

Appendix

Table 8: Compute Resource Usage for DFT XC Functional Parameter Fitting.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$2.2\times 10^{-3}\pm 8.2\times 10^{-4}$	$8.7\times 10^{0}\pm 0.0$
RMSprop	$2.0\times 10^{-3}\pm 5.0\times 10^{-5}$	$8.7\times 10^{0}\pm 0.0$
Adam	$2.4\times 10^{-3}\pm 1.7\times 10^{-4}$	$8.7\times 10^{0}\pm 0.0$
AdamW	$2.6\times 10^{-3}\pm 2.3\times 10^{-4}$	$8.7\times 10^{0}\pm 0.0$
LBFGS	$6.4\times 10^{-1}\pm 8.4\times 10^{-2}$	$8.9\times 10^{0}\pm 9.2\times 10^{-4}$
KFAC	$2.8\times 10^{-3}\pm 2.0\times 10^{-4}$	$8.7\times 10^{0}\pm 0.0$

Appendix

Table 9: Compute Resource Usage for Tire Magic Formula Parameter Fitting.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$9.1\times 10^{-2}\pm 4.0\times 10^{-4}$	$9.1\times 10^{0}\pm 0.0$
RMSprop	$8.6\times 10^{-2}\pm 1.4\times 10^{-3}$	$9.1\times 10^{0}\pm 0.0$
Adam	$8.6\times 10^{-2}\pm 8.1\times 10^{-4}$	$9.1\times 10^{0}\pm 0.0$
AdamW	$8.7\times 10^{-2}\pm 7.8\times 10^{-4}$	$9.1\times 10^{0}\pm 0.0$
LBFGS	$6.4\times 10^{-1}\pm 2.4\times 10^{-2}$	$9.2\times 10^{0}\pm 6.3\times 10^{-3}$
KFAC	$9.3\times 10^{-2}\pm 1.7\times 10^{-3}$	$9.1\times 10^{0}\pm 0.0$

Appendix

Table 10: Compute Resource Usage for Battery Thevenin Model Parameter Fitting.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$6.8\times 10^{-1}\pm 1.6\times 10^{-3}$	$1.8\times 10^{1}\pm 0.0$
RMSprop	$6.9\times 10^{-1}\pm 9.0\times 10^{-3}$	$1.8\times 10^{1}\pm 0.0$
Adam	$8.1\times 10^{-1}\pm 9.8\times 10^{-2}$	$1.8\times 10^{1}\pm 0.0$
AdamW	$1.2\times 10^{0}\pm 5.5\times 10^{-3}$	$1.8\times 10^{1}\pm 0.0$
LBFGS	$2.0\times 10^{0}\pm 2.1\times 10^{-2}$	$1.8\times 10^{1}\pm 4.8\times 10^{-3}$
KFAC	$1.1\times 10^{0}\pm 5.7\times 10^{-3}$	$1.8\times 10^{1}\pm 0.0$

Appendix

Table 11: Compute Resource Usage for Tight Binding Parameter Fitting.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$6.8\times 10^{-3}\pm 1.1\times 10^{-4}$	$1.2\times 10^{1}\pm 0.0$
RMSprop	$6.8\times 10^{-3}\pm 7.0\times 10^{-5}$	$1.2\times 10^{1}\pm 0.0$
Adam	$7.2\times 10^{-3}\pm 1.7\times 10^{-4}$	$1.2\times 10^{1}\pm 0.0$
AdamW	$7.1\times 10^{-3}\pm 8.9\times 10^{-5}$	$1.2\times 10^{1}\pm 0.0$
LBFGS	$3.2\times 10^{0}\pm 1.6\times 10^{-1}$	$1.2\times 10^{1}\pm 0.0$
KFAC	$2.5\times 10^{-2}\pm 4.1\times 10^{-4}$	$1.2\times 10^{1}\pm 0.0$

Appendix

Table 12: Compute Resource Usage for Gaussian Splatting 2D Reconstruction.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$1.0\times 10^{-3}\pm 8.3\times 10^{-6}$	$8.7\times 10^{0}\pm 0.0$
RMSprop	$1.3\times 10^{-3}\pm 6.3\times 10^{-5}$	$8.7\times 10^{0}\pm 0.0$
Adam	$3.5\times 10^{-3}\pm 2.7\times 10^{-3}$	$8.7\times 10^{0}\pm 0.0$
AdamW	$1.8\times 10^{-3}\pm 4.0\times 10^{-5}$	$8.7\times 10^{0}\pm 0.0$
LBFGS	$6.4\times 10^{-2}\pm 2.5\times 10^{-4}$	$8.7\times 10^{0}\pm 0.0$
KFAC	$1.8\times 10^{-3}\pm 1.2\times 10^{-4}$	$8.7\times 10^{0}\pm 0.0$

Appendix

Table 13: Compute Resource Usage for Projectile Motion Range Matching.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$4.9\times 10^{-1}\pm 1.5\times 10^{-2}$	$9.3\times 10^{0}\pm 0.0$
RMSprop	$4.8\times 10^{-1}\pm 7.0\times 10^{-3}$	$9.3\times 10^{0}\pm 0.0$
Adam	$5.2\times 10^{-1}\pm 3.1\times 10^{-3}$	$9.3\times 10^{0}\pm 0.0$
AdamW	$5.1\times 10^{-1}\pm 2.5\times 10^{-3}$	$9.3\times 10^{0}\pm 0.0$
LBFGS	$6.6\times 10^{-1}\pm 2.8\times 10^{-2}$	$9.4\times 10^{0}\pm 5.0\times 10^{-3}$
KFAC	$5.2\times 10^{-1}\pm 1.9\times 10^{-2}$	$9.3\times 10^{0}\pm 0.0$

Appendix

Table 14: Compute Resource Usage for Fluid Mechanics Shape Optimization.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$1.5\times 10^{-2}\pm 3.4\times 10^{-4}$	$3.8\times 10^{1}\pm 0.0$
RMSprop	$1.4\times 10^{-2}\pm 8.5\times 10^{-5}$	$3.8\times 10^{1}\pm 0.0$
Adam	$1.4\times 10^{-2}\pm 7.3\times 10^{-5}$	$3.8\times 10^{1}\pm 0.0$
AdamW	$1.4\times 10^{-2}\pm 2.2\times 10^{-4}$	$3.8\times 10^{1}\pm 0.0$
LBFGS	$5.0\times 10^{-1}\pm 4.1\times 10^{-2}$	$3.8\times 10^{1}\pm 4.9\times 10^{-3}$
KFAC	$3.1\times 10^{-2}\pm 3.1\times 10^{-4}$	$3.8\times 10^{1}\pm 0.0$

Appendix

Table 15: Compute Resource Usage for Atmospheric Scattering Chromaticity Maximization.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$3.6\times 10^{-1}\pm 5.9\times 10^{-4}$	$4.4\times 10^{1}\pm 0.0$
RMSprop	$3.4\times 10^{-1}\pm 5.6\times 10^{-4}$	$4.4\times 10^{1}\pm 0.0$
Adam	$3.4\times 10^{-1}\pm 1.1\times 10^{-3}$	$4.4\times 10^{1}\pm 0.0$
AdamW	$3.4\times 10^{-1}\pm 1.4\times 10^{-3}$	$4.4\times 10^{1}\pm 0.0$
LBFGS	$2.0\times 10^{0}\pm 1.6\times 10^{-1}$	$4.4\times 10^{1}\pm 2.2\times 10^{-2}$
KFAC	$3.5\times 10^{-1}\pm 2.1\times 10^{-3}$	$4.9\times 10^{1}\pm 0.0$

Appendix

Table 16: Compute Resource Usage for Elastic Mechanics Topology Optimization.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$1.6\times 10^{0}\pm 1.4\times 10^{-2}$	$1.1\times 10^{1}\pm 0.0$
RMSprop	$1.7\times 10^{0}\pm 1.1\times 10^{-2}$	$1.1\times 10^{1}\pm 0.0$
Adam	$1.6\times 10^{0}\pm 7.3\times 10^{-3}$	$1.1\times 10^{1}\pm 0.0$
AdamW	$1.5\times 10^{0}\pm 5.0\times 10^{-3}$	$1.1\times 10^{1}\pm 0.0$
LBFGS	$5.0\times 10^{0}\pm 4.9\times 10^{0}$	$1.1\times 10^{1}\pm 2.1\times 10^{-2}$
KFAC	$2.1\times 10^{0}\pm 5.1\times 10^{-3}$	$1.3\times 10^{1}\pm 0.0$

Appendix

Table 17: Compute Resource Usage for Flight Dynamics Range Maximization.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$2.9\times 10^{0}\pm 5.4\times 10^{-2}$	$1.9\times 10^{1}\pm 0.0$
RMSprop	$2.8\times 10^{0}\pm 2.2\times 10^{-2}$	$1.9\times 10^{1}\pm 0.0$
Adam	$2.6\times 10^{0}\pm 3.5\times 10^{-2}$	$1.9\times 10^{1}\pm 0.0$
AdamW	$2.8\times 10^{0}\pm 1.8\times 10^{-1}$	$1.9\times 10^{1}\pm 0.0$
LBFGS	$1.3\times 10^{1}\pm 7.2\times 10^{0}$	$1.9\times 10^{1}\pm 1.3\times 10^{-2}$
KFAC	$3.6\times 10^{0}\pm 5.2\times 10^{-3}$	$1.9\times 10^{1}\pm 0.0$

Appendix

Table 18: Compute Resource Usage for Rigid-Body Robot Setpoint.

Optimizer	Step wall (s), mean $\pm$ std	Peak GPU (MB), mean $\pm$ std
SGD	$1.4\times 10^{0}\pm 1.1\times 10^{-2}$	$9.3\times 10^{0}\pm 0.0$
RMSprop	$1.4\times 10^{0}\pm 2.3\times 10^{-3}$	$9.3\times 10^{0}\pm 0.0$
Adam	$1.4\times 10^{0}\pm 3.8\times 10^{-3}$	$9.3\times 10^{0}\pm 0.0$
AdamW	$1.4\times 10^{0}\pm 1.5\times 10^{-2}$	$9.3\times 10^{0}\pm 0.0$
LBFGS	$3.9\times 10^{0}\pm 6.5\times 10^{-1}$	$9.4\times 10^{0}\pm 2.3\times 10^{-2}$
KFAC	$7.5\times 10^{-1}\pm 2.0\times 10^{-3}$	$1.1\times 10^{1}\pm 0.0$

Appendix

Table 19: Compute Resource Usage for Mass-Spring Robot Reach.

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Mosaic: A Benchmark Suite for Differentiable Physics Solvers

Andrin Rehmann, Heiko Zimmermann, Dion Häfner

Pasteur Labs, Brooklyn, NY, USA

Automatic Differentiation PDE Solving Benchmark Design Differentiable Programming Differentiable Physics Partial Differential Equations

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Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, computational cost, gradient accuracy, and numerical conditioning vary widely across solvers and are discoverable only by trial and error. We introduce Mosaic, an extensible benchmarking framework for differentiable PDE solvers that standardizes access to solver gradients. Each solver is packaged as a containerized component (Tesseract) exposing a uniform gradient API regardless of language or automatic differentiation (AD) strategy, enabling researchers to evaluate, compare, and build on non-trivial physical solvers. Our evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer demonstrates that the benchmark surfaces practically relevant differences: order-of-magnitude variation in computational cost and Jacobian conditioning, alongside structural incompatibilities that eliminate solvers from realistic tasks entirely. Despite this variation, all solvers that produce gradients converge to similar optima, indicating that the practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy alone. Mosaic is open-source and available at https://github.com/pasteurlabs/mosaic.

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