# An Extensible Benchmark Suite for Learning to Simulate Physical Systems

arXiv: https://arxiv.org/abs/2108.07799 · HTML: https://mlsanity.com/paper/2108.07799/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 63 references that are in the map.
- First posted on arXiv 9 Aug 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python](https://mlsanity.com/paper/1907.10121/) (2019)
- [Stress Field Prediction in Cantilevered Structures Using Convolutional Neural Networks](https://mlsanity.com/paper/1808.08914/) (2018)
- [DeepCFD: Efficient Steady-State Laminar Flow Approximation with Deep Convolutional Neural Networks](https://mlsanity.com/paper/2004.08826/) (2020)
- [Reconstruction of Simulation-Based Physical Field by Reconstruction Neural Network Method](https://mlsanity.com/paper/1805.00528/) (2018)

## Built on by (most cited first, 24 of 24)

- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions](https://mlsanity.com/paper/2212.07564/) (2022)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators](https://mlsanity.com/paper/2109.10793/) (2021)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learning](https://mlsanity.com/paper/2307.14623/) (2023)
- [Data Representativity for Machine Learning and AI Systems](https://mlsanity.com/paper/2203.04706/) (2022)
- [Elucidated Rolling Diffusion Models for Probabilistic Forecasting of Complex Dynamics](https://mlsanity.com/paper/2506.20024/) (2025)
- [Diffusion Models for Time Series Forecasting: A Survey](https://mlsanity.com/paper/2507.14507/) (2025)
- [DynaBench: A benchmark dataset for learning dynamical systems from low-resolution data](https://mlsanity.com/paper/2306.05805/) (2023)
- [MDBench: Benchmarking Data-Driven Methods for Model Discovery](https://mlsanity.com/paper/2509.20529/) (2025)
- [An extensible Benchmarking Graph-Mesh dataset for studying Steady-State Incompressible Navier-Stokes Equations](https://mlsanity.com/paper/2206.14709/) (2022)
- [Training Deep Surrogate Models with Large Scale Online Learning](https://mlsanity.com/paper/2306.16133/) (2023)
- [MetaSym: A Symplectic Meta-learning Framework for Physical Intelligence](https://mlsanity.com/paper/2502.16667/) (2025)
- [Learning to Predict Structural Vibrations](https://mlsanity.com/paper/2310.05469/) (2023)
- [Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes](https://mlsanity.com/paper/2310.07485/) (2023)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [SimbaML: Connecting Mechanistic Models and Machine Learning with Augmented Data](https://mlsanity.com/paper/2304.04000/) (2023)
- [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026)
- [First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems](https://mlsanity.com/paper/2606.11138/) (2026)
- [One-Step Generative Surrogate Models via Block-Triangular Joint Drifting](https://mlsanity.com/paper/2609.26435/) (2026)

## Abstract (arXiv)

Simulating physical systems is a core component of scientific computing, encompassing a wide range of physical domains and applications. Recently, there has been a surge in data-driven methods to complement traditional numerical simulations methods, motivated by the opportunity to reduce computational costs and/or learn new physical models leveraging access to large collections of data. However, the diversity of problem settings and applications has led to a plethora of approaches, each one evaluated on a different setup and with different evaluation metrics. We introduce a set of benchmark problems to take a step towards unified benchmarks and evaluation protocols. We propose four representative physical systems, as well as a collection of both widely used classical time integrators and representative data-driven methods (kernel-based, MLP, CNN, nearest neighbors). Our framework allows evaluating objectively and systematically the stability, accuracy, and computational efficiency of data-driven methods. Additionally, it is configurable to permit adjustments for accommodating other learning tasks and for establishing a foundation for future developments in machine learning for scientific computing.