cs.ROSep 15, 2026
SaveEmbedding Physics Priors in Robot Learning: A Survey
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.
Abstract
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.
| Survey Paper | Physics-Guided Inputs / Data | Physics-Encoded Architectures | Physics-Informed Losses | Machine Learning Models 1 | Physics Priors in Robotics 2 |
| Shlezinger et al. (2023) | ✓ | Multi-Layer Perceptron (MLP) , Recurrent Neural Network (RNN) , Variational Autoencoder (VAE) , Convolutional Neural Network (CNN) , Generative Adversarial Network (GAN) | |||
| Cuomo et al. (2022) | ✓ | MLP , RNN , Long Short-Term Memory Network (LSTM) , Bayesian Neural Network (BNN) , CNN , GAN , NO | |||
| Goswami et al. (2023) | ✓ | NO | |||
| Faroughi et al. (2024) | ✓ | ✓ | ✓ | MLP , RNN , LSTM , GAN , CNN , VAE , NO | |
| Karniadakis et al. (2021) | ✓ | ✓ | ✓ | MLP , RNN , BNN , CNN , GAN , GPR , NO | |
| Geist and Trimpe (2021) | ✓ | MLP , GPR , analytical parametric NNs , analytical residual models | ✓(16/108) 2 |
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.
Figure 5: Sub-categories of physics-encoded robot learning architectures (Sec. 3 ).
| Paper | Application | Robot type | Additional physical principles | Method |
| Dynamics Learning | ||||
| Cranmer et al. (2020) ; Lu et al. (2022) | System Identification (SysID) | Pendulum | LNN | |
| Lahoud et al. (2025) | Parallel robot | Sparsity from decoupled translational and rotational inertia | DeLaN | |
| Schulze et al. (2026) | Quadrupeds / humanoids | Branch-induced sparsity, full physical consistency of the spatial inertia | ||
| Wu et al. (2024) | Manipulator | |||
| Lahoud et al. (2024) | Viscous and non-symmetrical Coulomb friction, motor rotor inertia | |||
Table 2: Summary of the reviewed papers on Lagrangian and Hamiltonian learning (Sec. 3.1 - 3.2 ), grouped by application category.
| Paper | Application | Robot | Physical Principles in Archit. Layers / Connections / Design | Physical Constraints |
| Dynamics Learning | ||||
| Da Lio et al. (2020a) ; James et al. (2020) | Longitudinal dynamics learning | Vehicle | Force superposition; drag law; discrete gear-dependent dynamics | ✗ |
| Piccinini et al. (2025b) | Longitudinal dynamics learning / friction estimation | Vehicle | 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 |
| HNN /port-Hamiltonian | Hamiltonian structure, symplectic geometry, passivity | Systems governed by energy exchange / dissipation / passivity | Energy-based interpretation; explicit treatment of dissipation and interconnection | Requires generalized momenta or their reconstruction from observable states; limited validation on high- DoF robots |
| LNN | Euler–Lagrange equations | General Lagrangian systems | Greater flexibility than DeLaN while retaining energy-based interpretation | Weaker structural guarantees and reduced interpretability than DeLaN |
| MSNN | Physical principles in architectural layers / connections / constraints | Known dependencies among physical quantities; force decomposition; bounds on physical quantities | Application-specific architecture design; interpretable trained sub-modules | Substantial domain expertise required; limited generality across robot classes; no prescribed design rules |
| Topology learning, SINDy -like | Physics-based candidate operators designed from first principles | Discovering governing equations from data | Compact and interpretable models | Basis candidate design remains manual; limited scalability to contact-rich systems |
| Neural operators | Physics-aware lifting functions and observables | Complex nonlinear dynamics | Enables linear control methods for underlying nonlinear dynamics | Application-specific lifting design; finite lifting limits exactness |
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 | Vehicle | Residual penalties w.r.t. analytical vehicle models (single-track / tractor-trailer) and physics-based regularization terms |
| Beaber et al. (2024) ; Wang et al. (2024b) | Soft-robot deformation surrogate modeling | Soft robot | 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.
More in this topic
Robot Learning
Robot learning develops methods for robots to acquire skills and adapt from demonstrations, interaction, data, and physical models.
See all 321 papers