Embedding Physics Priors in Robot Learning: A Survey
Authors: Mattia Piccinini, Lucas Schulze, Alice Plebe, Matteo Saveriano, Thomas Beckers, Yuan Gao, Oleg Arenz, Baha Zarrouki, Dingrui Wang, Finn Rasmus Schäfer, Jan Peters, Johannes Betz, Gastone Pietro Rosati Papini
Organizations: Professorship of Autonomous Vehicle Systems, Technical University of Munich, 85748 Garching, Germany; Munich Institute of Robotics and Machine Intelligence (MIRMI). · Department of Computer Science, Technical University of Darmstadt, Germany. · Department of Industrial Engineering, University of Trento, 38123 Trento, Italy. · Institute of Measurement, Control and Microtechnology, Ulm University, 89081 Ulm, Germany.
The rapid progress of artificial intelligence is reshaping robotics and accelerating the adoption of learning-based approaches. While purely data-driven methods have achieved remarkable success in computer vision and natural language processing, robotics remains constrained by limited data, complex real-world interactions, and the need for reliable operation. These challenges have motivated the exploration of physics-embedded robot learning, which embeds physics priors into learning algorithms. By encoding the underlying physical laws and constraints, physics priors can complement limited data with robotics-specific inductive biases, potentially improving generalization, interpretability, and sample efficiency. However, the literature on physics-embedded robot learning remains fragmented across terminology, methodologies, and application domains, making it difficult to assess this growing body of work. This survey reviews physics-embedded robot learning across a broad range of physics priors, robotics applications, and machine learning models, from single-layer perceptrons to generative foundation models. We adopt a unified taxonomy that classifies existing approaches according to their physics embedding: physics-guided inputs, data, and representations; physics-encoded model architectures; and physics-informed training loss functions. Building on this taxonomy, we review methods for robot dynamics learning, trajectory planning, prediction, control, and estimation, together with the corresponding open-source software ecosystem. We identify key open challenges, and outline promising future research directions. Overall, we argue that physics priors provide a particularly relevant robotics-specific inductive bias, complementing rather than replacing data-driven learning, and paving the way toward more generalizable, data-efficient, and trustworthy robotic systems.
Figures & tables
Figure 1: Different levels of embedding physics priors in robot learning: (a) physics-guided inputs, data, and representations; (b) physics-encoded architectures; (c) physics-informed loss functions. While most existing works focus on only one of these levels, we argue that the joint use of complementary routes may enable richer exploitation of prior physical knowledge, but systematic comparisons remain limited.
Figure 2: Distribution of the reviewed papers by application (left) and robot class (right), grouped by the three levels of physics embedding. In the robot class plot, “canonical mechanical systems” include pendulums, cart-poles, and other standard mechanical systems used for benchmarking. Note that some papers are counted in more than one application or robot class, as they address multiple tasks or robot platforms. Some papers also employ two physics embedding routes.
Table 1: Analysis of the most relevant surveys on embedding physics or model-based priors into learning methods. Our survey is the first to cover physics-guided inputs/data, physics-encoded architectures, and physics-informed losses in the context of robot learning, including a broad range of ML models, physics priors, and robotics applications.
Figure 3: Lifecycle of physics priors in robot learning. Physics-guided components transform or enrich input features and data, either during data curation or as pre-processing modules that remain active at training and inference time. Physics-encoded priors are embedded in the model architecture, and remain active during training and inference. Physics-informed components embed physical knowledge via the training loss, and are active only during training.
Figure 4: Decision flow to classify physics-embedded robot learning approaches. Methods that embed only generic mathematical structure, or that are not applied to robotic systems, do not satisfy the first criterion and fall outside the scope of this survey. The three categories (physics-guided, physics-encoded, physics-informed) are not mutually exclusive: multiple labels may be assigned by sequentially evaluating all criteria.
Newtonian longitudinal dynamic laws; dependency of the dynamics on past input history
✗
Chrosniak et al. (2024) ; Kolluri et al. (2025) ; Fang and Yu (2025)
Vehicle dynamics learning / estimation
Vehicle / race car
Single-track vehicle parameterization
Parameters constrained to remain physically meaningful
Kamp et al. (2023)
Suspension dynamics learning
Vehicle
Spring-damper dynamics
Regularization penalizing non-zero damper force at zero velocity
Mungiello et al. (2026)
Coupled longitudinal-lateral dynamics learning
Race car
Steady-state, transient, and coupling decomposition
✗
Table 3: Summary of the reviewed papers on model-structured learning approaches (Sec. 3.3 ), grouped by application category.
Paper
Application
Robot
Topology-Learning / Physics-Encoding Principles
Dynamics Learning
Papageorgiou et al. (2024)
Dynamics learning
Soft robot
SINDy library from kinematic/dynamic principles, including trigonometric terms for rotational effects
Liu et al. (2024a)
Dynamics learning
Unmanned Marine Vehicle (UMV)
Physics-inspired SINDy candidate functions derived from Newtonian and hydrodynamic terms
Buriani et al. (2026)
Dynamics learning
Jumping quadruped
Reduced-order SINDy in a physically interpretable latent space
Omar et al. (2024)
Dynamics learning with friction and payload effects
Manipulator
Euler-Lagrange-informed SINDy library with inequality-constrained sparse identification
Bhattacharya et al. (2020)
Dynamics equation discovery
Soft robotic esophagus
Reduced-order modeling combined with SINDy for sparse differential equation discovery
Table 4: Summary of the reviewed papers on topology learning with physics-encoded architectures (Sec. 3.6 ).
Paper
Neural Operator
Robot
P.E.
P.I.
P.G.
How Physics is Incorporated
Dynamics Learning
Zhang et al. (2026)
Koopman
Vehicle
✓
Acceleration-consistency physics loss
Ristich et al. (2025)
Soft robotic arm
✓
PDE-residual split-operator training
Shi and Karydis (2022)
Wheeled robot / 2-link arm / soft robotic leg
✓
Topology-derived analytical lifting dictionary
Wang et al. (2025b)
Kolmogorov-Arnold Network (KAN)
Manipulator
✓
Interpretable friction-law parameterization
Tan et al. (2025)
Deep Operator Network (DeepONet)
Vehicle
✓
✓
Model-based latent state mapping + physics loss
Table 5: Summary of the reviewed neural-operator papers for robotics and related robotic systems. Checkmarks indicate whether a paper embeds physics through operator architecture design (Physics-Encoded, P.E., Sec. 3.7 ), physics-based training loss functions (Physics-Informed, P.I., Sec. 4.2 ), and/or physically structured inputs, states or data (Physics-Guided, P.G., Sec. 5.7 ).
Paper
Application
Generative Model
P.E.
P.I.
P.G.
Robot
How Physics is Incorporated
Symmetry-Aware Architectures
Yang et al. (2024)
Control
Diffusion
✓
Mobile manipulator
SIM(3)-equivariant architecture for scaling/translation robustness
Wang et al. (2024a)
✓
Manipulator
SO(2)/T(3)-equivariant denoising policy for sample-efficient learning
Seo et al. (2025)
✓
SE(3)-equivariant pipeline from perception to action generation
Tie et al. (2025)
Trajectory planning and control
Diffusion
✓
Manipulator
Trajectory-level SE(3) equivariance with Markov-kernel denoising
Zhu et al. (2025)
✓
SE(3)-equivariant spherical-Fourier conditioning and denoising
Table 6: Summary of generative-model approaches, reviewed throughout this survey, grouped by how physics is incorporated. Checkmarks indicate whether a paper embeds physics through architecture design (Physics-Encoded, P.E.), physics-based training loss functions (Physics-Informed, P.I.), and/or physically structured inputs, data or representations (Physics-Guided, P.G.).
Family
Main Embedded Priors
When to Use
Main Strengths
Main Limitations
DeLaN
Energy conservation; energy decomposition; positive-definite inertia matrix
Non-conservative forces are known or negligible
Interpretable physical quantities; energy, inverse, and forward dynamics from one learned model; reusable in model-based control
Assumes collocated coordinates; dissipation and contact terms require separate models
Table 7: Comparison of the main physics-encoded architecture families reviewed in Sec. 3 . The table highlights the design trade-offs associated with different ways of embedding physics into learning architectures, including the encoded physics prior, the scenarios in which each family is most suitable, and the corresponding strengths and limitations. The comparison refers to architectural properties rather than measured performance.
Paper
Application
Robot
Physical Principles Embedded in the Loss Function
Dynamics Learning
Sun et al. (2024) ; Lim et al. (2024) ; Zeipel et al. (2024)
Vehicle dynamics and energy-related dynamics prediction
Continuum-mechanics constraints and minimum-potential-energy principles in the training objective
Bensch et al. (2024)
Continuum robot shape learning
Continuum robot
Cosserat-rod-inspired PDE residual losses with boundary-condition consistency
Serrano et al. (2024) ; Wang et al. (2025a)
Aerial system dynamics modeling under disturbances
Quadrotor / multirotor with slung load
Discretized first-principles dynamics consistency, Newton–Euler-informed residuals, and wind-aware physics penalties
Trajectory Planning
Table 8: Summary of the reviewed papers using physics-informed loss functions (Sec. 4.1 ), grouped by application category.
Paper
Category
Robot
How Physics is Incorporated in Inputs / Data / Representations
Dynamics Learning
De Groote et al. (2021)
Physical models as structured NN inputs
Slider-crank mechanism
Physics-based model computes displacement and velocity features as MLP inputs
Lahoud et al. (2024)
Manipulator
Selection of physically meaningful inputs and Fourier-series excitation trajectory design, matched to the robot’s natural modes
Robinson et al. (2022)
Mechanical oscillator
Features computed from physics-based equations injected into MLP hidden layers
Wu et al. (2025)
Lower-limb prosthetics
Discrete Cosine Transform (DCT) encoding of human kinematics frequency content
Trajectory Planning
Table 9: Summary of the reviewed papers on physics-guided inputs, data, and representations (Sec. 5 ), grouped by application category. Papers using generative models (Sec. 5.5 ) are collected in Table 6 , while the ones about physics-guided neural operators (Sec. 5.7 ) are reported in Table 5 .
Framework
Language (Backend)
Optimizers
P.E.
P.I.
License
Sim
DL
Ctrl
Hardware
URL
Software for Model-Structured & Hybrid Physics-Learning Architectures
NeuroMANCER ( Drgona et al., 2023 ; Drgona et al., 2026 )
Python (PyTorch)
Wrapper
✓
✓
BSD 3-Clause
✓
✓
CPU/GPU
https://github.com/pnnl/neuromancer
nnodely ( Rosati Papini, 2026 )
Python (PyTorch/JAX/TF/Keras)
Wrapper
✓
✓
MIT
✓
✓
CPU/GPU
https://github.com/tonegas/nnodely
Software for Neural Ordinary Differential Equations
torchdiffeq ( Chen, 2026 )
Python (PyTorch)
Wrapper
✓
MIT
✓
✓
CPU/GPU
https://github.com/rtqichen/torchdiffeq
torchdyn ( Poli et al., 2020 ; DiffEqML, 2026 )
Python (PyTorch)
Wrapper
✓
Apache-2.0
✓
✓
∼
CPU/GPU
https://github.com/DiffEqML/torchdyn
Table 10: Open-source software frameworks for embedding physics priors in robot learning. Checkmarks indicate whether a framework supports physics-encoded architectures (P.E.), physics-informed loss functions (P.I.), and the target tasks of simulation (Sim), dynamics learning/estimation (DL), and control (Ctrl). The Hardware column reports deployment options, while the last column links to the URL of the open-source software page.
Real robot data, rich across tasks and morphologies, is often treated as the bottleneck for training robot learning models, including embodied foundation models that aim for zero-shot physical transfer. In this paper, we investigate the degree to which such models can acquire necessary structure using data from systems without embodiment. To this end, we test if inexpensive, procedurally generated synthetic data can yield useful priors before a model observes any real-world behavior. We study this through robot action prediction, a natural pretraining objective and a direct test of whether structure learned from arbitrary synthetic tasks transfers to embodied settings. We introduce three data generation pipelines that model mathematical abstractions of robotic control, and use them as pretraining priors for a Transformer. We demonstrate that a model trained purely on this non-robotic data attains non-trivial zero-shot prediction performance on unseen physical robot tasks via an in-context learning mechanism, and that our specialized control priors notably improve upon generic causal priors. We also compare against a compute-matched baseline trained on real robot data, and find that it outperforms our model but only on the datasets it was trained on. These findings suggest that procedurally generated, mathematical abstractions can yield useful robotic priors, offering a potential complement to expensive embodied data collection.
Mateusz Odrowaz-Sypniewski, Michael Amir, Jasmine Bayrooti +2
Department of Computer Science and Technology, University of Cambridge, UK
Living organisms exhibit extraordinary adaptability to unseen environments through their intrinsic physical structures and lifelong feedback-driven learning. Endowing robots with comparable generalization is critical for reliable operation in the real world. While recent approaches attempt to improve generalization by scaling training data, such strategies remain impractical for robotics, where collecting real-world demonstrations at the scale of large language models is prohibitively costly and slow. Contrary to this reliance on massive datasets, we show that robots can generalize effectively under dynamics uncertainties even with limited training data by leveraging a feedback mechanism, namely PhyFilter, that corrects learning outputs with physics-filtered learning residuals. PhyFilter operates as a lightweight, model-agnostic module whose parameters can be automatically optimized through an auto-learning algorithm, eliminating manual tuning and enabling seamless integration with diverse robot policies. We validate PhyFilter across four representative robotic systems, demonstrating that it enables quadruped robots to generalize to unseen terrains, payload variations, and speed ranges; drones to flight under unseen wind disturbances; aerial manipulators to achieve centimeter-level in-air capture despite wind and mass uncertainties; and acceleration differentiators to remain robust with distribution shift. These results show that physics-filtered feedback can serve as a powerful alternative to massive data scaling.
Jindou Jia, Shixuan Han, Meng Wang +8
Nanyang Technological University · Beihang University · National University of Singapore +1
A robot should be able to learn through experiments how unfamiliar objects behave and interact, then plan with that knowledge. It need not start from scratch: physics engines supply knowledge of motion and contact, but can omit entire mechanisms, such as glue curing, water heating, or wind. We present EMPIRIC, an agent that learns a residual world model: a physics engine extended with code for the missing mechanisms. The learned programs can introduce new forces, constraints, and hidden state, and Bayesian inference estimates their parameters and states from noisy observations. The resulting model lets the agent predict the outcomes of actions, choose informative experiments, and revise its hypotheses when predictions fail. Across five simulated domains, EMPIRIC learns interpretable, reusable models, and solves more tasks with fewer environment interactions than all three baselines. On a physical robot, it learns wind forces and domino masses to solve a manipulation task. Website and code: https://yichao-liang.github.io/empiric
Yichao Liang, Amber Li, Dat Nguyen +13
Basis Research Institute · University of Cambridge · Carnegie Mellon University +6