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Embedding Physics Priors in Robot Learning: A Survey

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

arXiv:2609.22319v1cs.RO

TL;DR

Robotics faces costly data collection, complex physical interactions, and demanding reliability requirements, while research on physics-embedded learning remains fragmented. This survey organizes the field through a unified taxonomy of where physics priors enter learning methods. It finds that physics-encoded architectures make up 71% of reviewed methods, while arguing that priors should complement data-driven learning rather than replace it.

  • Problem

    Robotics faces expensive data collection, complex real-world interactions, and reliability demands, while the survey identifies fragmented terminology and methods in physics-embedded learning.

    Source: §1 · pp. 1–2

  • Method

    The survey reviews physics-embedded robot learning using a taxonomy of physics-guided inputs, data, and representations, physics-encoded architectures, and physics-informed losses.

    Source: §1 · p. 2

  • Results

    Physics-encoded architectures account for 71% of reviewed methods, and the survey finds no universally optimal embedding strategy.

    Source: §9 · p. 33

  • Takeaways & Limitations

    Physics priors are robotics-specific inductive biases that should complement data-driven learning, with embedding choices depending on prior fidelity, available data, and closed-loop needs.

    Source: §9 · p. 33

    Physics-informed laws are enforced only as soft training constraints, so low residual losses do not guarantee exact physical consistency at inference.

    Source: §4.1 · p. 22

Abstract

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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.

1 Introduction

Robotics’ costly data collection, complex physical interactions, and reliability requirements motivate physics-embedded learning as a complement to data-driven methods. The survey unifies fragmented work with a taxonomy, reviews its scope and related literature, and traces the historical development of physics priors in robot learning.

1 Introduction

  • Robotics combines expensive data collection with high-dimensional nonlinear dynamics, underactuation, and hybrid contact, complicating safe generalization to unseen scenarios.The survey frames physics priors as a robotics-specific inductive bias that complements rather than replaces data-driven learning, while noting a trade-off with model expressiveness.Source: §1 · pp. 1–2

1.1 Scope of the Considered Literature

  • The survey covers physics priors, learning models, and robotic applications, grouping embedding routes into physics-guided, physics-encoded, and physics-informed categories.Its scope includes five families of physical priors, models ranging from neural networks to generative models, and dynamics learning, planning and prediction, control, and estimation.Source: §1.1 · pp. 2–3

1.2 Related Surveys

  • The survey distinguishes its broad robotics coverage from earlier reviews that address other domains or narrower robotic topics.For example, a prior robotics survey was limited to rigid-body dynamics and did not cover control, trajectory planning, or state estimation.Source: §1.2 · p. 3, Table 1 · p. 4

1.3 Contributions

  • The survey contributes a structured review, a three-route taxonomy, application and software overviews, and an analysis of challenges and future directions.The review spans learning models from single-layer perceptrons to generative models and a broad range of physics priors.Source: §1.3 · p. 4

1.4 Historical Perspective: From Analytical Models to Physics-Embedded Learning

  • Physics-embedded robot learning developed alongside analytical and data-driven approaches, evolving toward physics-guided, physics-encoded, and physics-informed methods.Earlier examples incorporated rigid-body or Lagrangian structure into learning models and network topologies.Source: §1.4 · p. 5

2 Physics-Guided, Physics-Encoded, and Physics-Informed Robot Learning: Overview and Organization

The survey classifies physics-embedded learning by where physical knowledge enters the learning pipeline. Its categories can apply at different levels and can overlap within a single method.

Taxonomy

  • Physics-guided methods transform or enrich inputs, data, or representations; physics-encoded methods place physical knowledge in architectures; physics-informed methods add it to training objectives.These categories define physics-embedded learning as a way to complement data with available physical priors.Source: §2 · p. 6
  • The taxonomy can classify an entire pipeline, an individual model, or a submodule within a larger architecture..Source: §2 · p. 6

Lifecycle of Physics Priors

  • Physics-guided components may act during data curation or persist through training and inference, while encoded architectures remain active at both training and inference.Physics-informed residual losses act during training, with their influence at inference carried only through learned parameters.Source: §2 · p. 6

Classification Flow

  • The categories are not mutually exclusive, so a method may receive multiple labels through sequential evaluation of the classification criteria.The same decision flow is implemented in the open-source repository for ongoing classification of new methods.Source: §2 · p. 6

3 Physics-Encoded Architectures

Physics-encoded architectures build physical principles into model structures to align inductive biases with robot dynamics, and the survey organizes this broad approach into six classes. Lagrangian methods illustrate how these structures encode mechanics while varying in generality and physical structure.

3 Physics-Encoded Architectures

  • 71% of surveyed methods use physics-encoded architectures, most often for manipulators and vehicles, with development accelerating markedly since 2020.Source: §3 · p. 6
  • The survey covers six architecture classes, from Lagrangian and Hamiltonian models to structured architectures, neural ODEs, hybrid models, topology learning, and neural operators.Source: §3 · p. 6

3.1 Lagrangian Learning Models

  • Lagrangian learning derives robot motion from energy-based formulations and Euler–Lagrange equations; DeLaN parameterizes inertia and potential energy while enforcing a positive-definite inertia matrix through Cholesky factorization.Source: §3.1 · p. 7, §3.1.1 · p. 7

3.1.1 Deep Lagrangian Networks (DeLaN):

  • DeLaN extensions incorporate sparsity, context adaptation, and learned non-conservative forces, supporting dynamics models for control and system identification.Source: §3.1.1 · p. 7
  • Gupta et al. (2020) reported that DeLaN extensions generalized better to unseen data than general-purpose MLPs.Source: §3.1.1 · pp. 7–8

3.1.2 Lagrangian Neural Networks (LNNs):

  • LNNs learn the Lagrangian directly rather than imposing DeLaN’s energy decomposition, allowing more general Lagrangians but giving up that structure.Source: §3.1.2 · p. 9, §3.1.3 · p. 9

3.1.3 Other Lagrangian Learning Methods:

  • Other Lagrangian methods use Gaussian-process kernels and structured RKHS models for tracking, dynamics identification, and uncertainty-aware exoskeleton control.Source: §3.1.3 · p. 9
  • Lagrangian methods’ explicit energy decomposition suits rigid-body systems but limits broader applicability, while validation remains concentrated on low-DoF benchmarks.Source: §3.1.3 · p. 9

3.2 Hamiltonian Learning Models

Hamiltonian learning represents mechanical systems through total energy as a function of generalized coordinates and momenta. The section reviews HNNs and port-Hamiltonian extensions, including their energy structure, applications, and limitations.

3.2 Hamiltonian Learning Models

  • Hamiltonian models describe a system’s total energy as a function of generalized coordinates q and momenta p, and relate it to the Lagrangian through a Legendre transformation.The Hamiltonian equations rewrite the Euler–Lagrange equations using generalized forces and the time derivatives of coordinates and momenta.Source: §3.2 · p. 9

3.2.1 Hamiltonian Neural Networks (HNNs):

  • For autonomous conservative systems, HNNs learn H(q, p) with an MLP and minimize residuals of the Hamiltonian equations, making learned dynamics energy-conserving by construction.The loss compares equation-derived dynamics with time derivatives available from training data; derivatives of the learned Hamiltonian are computed by automatic differentiation.Source: §3.2.1 · pp. 9–10
  • HNN applications range from pendulum and mass-spring dynamics to controlled systems, point-cloud-based tracking of a real wheeled robot, soft-robot tracking, and generated trajectories for manipulation planning.The cited methods include learned input maps for generalized forces, dissipation terms for non-conservative effects, and structured energy parameterizations.Source: §3.2.1 · p. 10
  • SE(3) HNN and LNN formulations enforce kinematic constraints with Lagrange multipliers, but redundant states and required constraint knowledge have limited applications to low-dimensional systems.The reviewed examples include pendulums, quadrotors, and wheeled robots.Source: §3.2.1 · p. 10

3.2.2 Port-Hamiltonian Learning

  • Port-Hamiltonian models explicitly represent energy storage, dissipation, and external energy exchange, supporting passivity-based control and applications across mechanical and other physical systems.Reviewed examples include trajectory tracking, vehicle and quadrotor modeling, underwater-vehicle system identification, and switching or distributed models for hopping and soft robots.Source: §3.2.2 · p. 10
  • Hamiltonian methods retain interpretability and energy consistency, but momentum is rarely measured directly, unstructured MLPs lose energy decomposition unless designed in, and long rollouts require structure-preserving integration.SE(3) formulations also require known kinematic constraints and have mainly been validated on low-DoF platforms.Source: §3.2.2 · p. 10

3.3 Model-Structured Learning Architectures

Model-structured learning embeds physics priors in model architectures through specialized layers, physical connections, and architectural constraints. The survey covers neural and Gaussian-process approaches across robotics applications and notes both reported benefits and design demands.

3.3 Model-Structured Learning Architectures

  • Model-structured learning embeds physics priors through specialized layers or internal connections, and through architectural constraints that enforce physical consistency; the term covers approaches beyond neural networks.The survey adopts the broader term to unify fragmented terminology; neural-network instances are called MSNNs.Source: §3.3 · p. 10
  • Vehicle MSNNs encode force contributions, drag dependence, control-input history, steering characteristics, and transient effects for dynamics learning, prediction, estimation, and control.Reviewed examples include longitudinal and lateral dynamics, handling-limit driving, automated parking, and lateral-velocity estimation.Source: §3.3 · pp. 10–11
  • Other robotic MSNNs structure human-motion prediction and manipulator dynamics by modeling interaction forces, friction, and inertial effects with tailored layers.Manipulator examples include viscous, Coulomb, and nonlinear Stribeck friction terms.Source: §3.3 · pp. 11–12
  • Structured neural architectures also model brake emissions, discontinuous contact dynamics, and manipulator inverse dynamics by embedding physical dependencies, contact geometry, or Newton–Euler equations.The Equation-Embedded NN reports competitive torque prediction and parameter identification with limited data, outperforming black-box baselines under varying friction conditions.Source: §3.3 · p. 12
  • Architectural constraints encode physical parameter ranges, non-negativity, spring and damper behavior, friction structure, positive-definite inertia, and multi-body symmetries or contact-force relations.The reviewed examples include vehicle, manipulator, motor-control, and multi-body dynamics models.Source: §3.3 · p. 12
  • Gaussian-process architectures embed nonholonomic geometry, dissipative friction structure, or kinematic-chain organization for constraint-aware dynamics learning and robot-arm torque prediction.The nonholonomic method is reported to rigorously certify constraint satisfaction.Source: §3.3 · p. 12
  • Compared with general-purpose learning models, reviewed MSNNs show improved sample efficiency, limited-data generalization, and interpretability, but require engineering expertise and careful physical-structure choices.Performance depends strongly on the selected physical principles and the design of architectural layers, connections, and constraints.Source: §3.3 · p. 12

3.4 Neural ODEs and Variational Integrator Networks

Neural ODEs provide continuous-time learning through differentiable ODE solvers, while variational integrator networks build physical structure into discrete-time dynamics. The section also reviews hybrid approaches that learn subsystems, residuals, or sensor-derived signals alongside physics-based models.

3.4 Neural ODEs and Variational Integrator Networks

  • Neural ODEs embed standard ODE solvers in a neural network computation graph and use adjoint sensitivity for backpropagation through the solver to learn continuous-time dynamics.The discrete ResNet update is presented as an Euler discretization of a continuous-time transformation.Source: §3.4 · p. 12
  • Variational Integrator Networks match discrete-time equations of motion and impose symplecticity, energy, and momentum conservation as hard architectural constraints.Their cited evaluations include mechanical systems with noisy observations and image-pixel inputs; forced variants incorporate control and damping, with applications to cartpole MPC, pendulum and quadrotor control, and vehicle motion prediction.Source: §3.4 · p. 13
  • Variational integrator networks preserve physical structure, but applying them to complex robotic systems and contact-rich dynamics remains an open challenge.This contrasts with Neural ODEs’ use of general-purpose integrators, which do not preserve physical systems’ geometric and energy properties.Source: §3.4 · p. 13

3.5 Hybrid Physics-Learning Architectures

  • Hybrid architectures learn selected dynamics components or correct physics-model outputs, keeping learned modules separate while retaining unchanged physics-based components.This modular approach is used when accurate subsystem parameterization is costly, time-consuming, or requires extensive expertise.Source: §3.5 · p. 13, §3.5.1 · p. 13

3.5.1 Learning Complex

  • Learned subsystems such as tire forces and complex thrust or aerodynamic effects are integrated with physics-based models for robotic dynamics, control, and estimation, including long-horizon hexacopter prediction from three minutes of data.The hexacopter model combines rigid-body dynamics with learned effects and uses a distance-aware diffusion term for uncertainty estimates outside the training distribution.Source: §3.5.1 · p. 13

3.5.2 Residual Learning:

  • Residual learning corrects discrepancies from unmodeled dynamics, parameter uncertainty, or simplifying assumptions, and can improve physics-model accuracy without specialized architectures or explicit physics constraints.Its performance still depends on the underlying model’s fidelity and the learning method’s expressive power; learned sensor preprocessing can also provide virtual measurements or reference signals to physics-based estimators and controllers.Source: §3.5.2 · pp. 13–14
  • Residuals may model dynamics or control parameters, with reviewed examples spanning vehicle and manipulator models, motors, pushed-object planning, and ballistic grasp-and-throw policies.The TossingBot policy combines an analytical ballistic model with learned compensation for effects including mass distribution, friction, aerodynamics, and grasp offset.Source: §3.5.2 · pp. 13–14

3.6 Physics-Encoded Topology Learning

The survey reviews physics-encoded approaches that build physical knowledge into learned topologies, operator models, and task-specific architectures. Across these approaches, priors shape model structure or representations, with application-specific design and scalability remaining important challenges.

3.6.1 NN-based Topology Learning:

  • Physics-encoded topology learning combines basic operators designed around system physics to discover interpretable, generalizable models while reducing manual architecture design and domain-knowledge requirements.Reviewed examples cover inverse dynamics, modular control, rigid-body interactions, and trajectory imitation.Source: §3.6.1 · pp. 14–15

3.6.2 Sparse Identification of Nonlinear Dynamical Systems (SINDy):

  • SINDy variants embed physics in candidate libraries and sparse regression to discover compact robot dynamics, but depend heavily on expert choices of library and coordinates and struggle with noisy derivatives, poor excitation, and contact-rich systems.Scaling to high-dimensional, interacting robots remains an open challenge.Source: §3.6.2 · pp. 15–16

3.7 Physics-Encoded Neural Operators

  • Physics-encoded neural operators, especially Koopman models, lift nonlinear robot dynamics into approximately linear spaces where physics-designed observables support learning and control.Application-specific lifting functions, rapidly growing lifted dimensions, and limited exploration beyond Koopman constrain transfer and scale.Source: §3.7 · p. 16

3.8 Other Types of Physics-Encoded Architectures

  • Task-specific architectures encode physical constraints into prediction, planning, and control, including feasible trajectories, boundary conditions, catching policies, and hopping control laws.Examples include race-car prediction, mobile-robot maneuvers, dynamic catching, and hopping transferred from simulation to a robot.Source: §3.8.1 · p. 17

3.8.1 Architectures for Robot Motion Planning:

  • Motion-planning architectures use task and physics structure to produce motion or time fields, sometimes replacing iterative optimization with a forward pass; their priors and guarantees remain application- or model-specific.Time-field methods largely address static collision avoidance, with limited treatment of dynamic obstacles and actuation constraints.Source: §3.8.1 · pp. 17–18

3.8.2 Bio-inspired Architectures:

  • Bio-inspired architectures embed inductive biases through action competition, structured latent representations, and predictive objectives, with reported benefits for generalization, perception, interpretability, and scene prediction.The survey identifies combining biological and physics-based priors as a promising direction.Source: §3.8.2 · p. 18

3.8.3 Symmetry-Aware Architectures:

  • Symmetry-aware architectures encode geometric equivariance in robot models and generative policies, and are reported to improve generalization and sample efficiency over non-equivariant baselines.Reviewed generative approaches impose scaling, translation, rotational, or SE(3) equivariance in diffusion models and their denoising pipelines.Source: §3.8.3 · pp. 18–19

3.9 Trade-offs Among Physics-Encoded Architectures

Physics-encoded architecture families suit different kinds and amounts of prior knowledge; no family dominates. Stronger priors can improve interpretability but demand accurate models and often additional measurements, while weaker priors offer flexibility with fewer structural guarantees.

3.9 Trade-offs Among Physics-Encoded Architectures

  • Architecture choice depends on available priors and the desired balance between structural bias and modeling flexibility, rather than a single universally best family.The comparison concerns architectural properties, not measured performance.Source: §3.9 · p. 19, Table 7 · p. 20
  • When governing equations are unknown but expected to have a compact basis representation, topology-learning and SINDy-like methods can recover them from data.Mechanics-based models suit settings with known energy structure, while HNNs and port-Hamiltonian models suit energy exchange, dissipation, and passivity.Source: §3.9 · p. 19
  • Neural operators are attractive when nonlinear dynamics should be controlled with linear tools, whereas NODEs and VINs target continuous-time learning and long-horizon prediction.Contact-rich dynamics and validation on high-DoF robots are gaps shared across architecture families.Source: §3.9 · p. 20
  • Stronger priors generally improve interpretability but require accurate physical models and often additional measurements; weaker priors offer greater flexibility but rely more on data and provide fewer structural guarantees.Manual design remains necessary for choices such as SINDy candidate bases and neural-operator lifting functions.Source: §3.9 · p. 20

4 Physics-Informed Loss Functions

The survey examines physics-informed losses that add physical laws as training constraints while retaining conventional model architectures. Reviewed applications span dynamics, estimation, control, planning, and generative models, while soft constraints and equation requirements limit guarantees and scope.

4 Physics-Informed Loss Functions

  • Physics-informed losses combine data-driven objectives with physical-law constraints, and aerial robotics accounts for 43% of reviewed physics-informed methods, largely because low-dimensional models can be embedded in training losses.A small training residual does not necessarily ensure exact satisfaction of governing equations at inference.Source: §4 · p. 20

4.1 Physics-Informed Neural Networks

  • PINNs impose governing-equation residuals as soft training constraints, supporting data-scarce learning and surrogate modeling across robotic dynamics, estimation, control, and planning.They can reduce reliance on labeled data and support flexible model classes, but accuracy depends on loss balancing and equation fidelity; closed differentiable equations are needed, and low residuals do not guarantee exact inference-time consistency.Source: §4.1 · pp. 20–22

4.2 Physics-Informed Neural Operators

  • Physics-informed neural operators combine equation-based losses with data or model constraints across vehicle, soft-robot, and quadrotor applications.Reviewed uses include operator learning from PDE residuals and embedding learned operators in model-predictive control.Source: §4.2 · p. 22

4.3 Other Types of Loss and Reward Functions

  • Other physics-informed losses and rewards encode contact consistency, stability, parameter admissibility, reachable sets, and physical coherence in generative models.Diffusion and video-world-model approaches use such objectives to improve feasibility or physical plausibility of generated outputs.Source: §4.3 · pp. 22–23

5 Physics-Guided Inputs, Data, and Representations

Physics-guided methods embed priors through inputs, data, and representations, often leaving standard model architectures and objectives unchanged. Across applications, they offer flexible ways to support learning and generation, but their guarantees and effectiveness depend on the source and fidelity of the physical guidance.

5 Physics-Guided Inputs, Data, and Representations

  • Physics-guided learning modifies inputs, data, or representations rather than model architecture or training objectives; it is comparatively easy to integrate but remains less mature.It comprises 16% of surveyed papers and has grown rapidly since 2022; generative and foundation-model approaches often use physics guidance for physical consistency.Source: §5 · p. 23

5.1 Physical Models as Structured Inputs to Learning Algorithms

  • Physics-based models and structured features supply physically meaningful inputs to learned estimators and controllers, with reported gains in sample efficiency and generalization.These approaches require domain knowledge, and their performance may be limited by how accurately the models capture complex dynamics.Source: §5.1 · p. 24

5.2 Physics-Guided Features & Training Data

  • Engineered features, informative trajectories, and physics-generated or corrected data support tasks from friction learning and fault detection to manipulation and terrain traversal.Data-centric methods address data scarcity, but their priors do not constrain predictions at inference; manual feature and excitation design and simulator fidelity remain limitations.Source: §5.2 · pp. 24–25

5.3 Geometric Learning

  • Manifold-aware representations and geometric processing preserve data structure for rotation, navigation, and manipulation tasks, with reported improvements in learning performance.Some methods project data to a tangent space for learning and map outputs back to the manifold; Riemannian flow matching instead constructs a geometrically consistent target vector field.Source: §5.3 · p. 25

5.4 Frequency-Domain Learning

  • Frequency-domain representations exploit physical properties such as human-motion bandwidth or gait periodicity to create compact descriptors for motion and prosthetics applications.Human-motion methods use a 10 Hz cutoff or low-pass bandwidth; the prosthetics method selects retained coefficients using a cumulative-energy criterion.Source: §5.4 · p. 25

5.5 Physically Consistent World Representations

  • Physics-guided world representations improve generated-video plausibility for planning, prediction, and control, but physical validity remains distinct from visual realism.Most approaches rely on external physics models or estimated parameters, and current work focuses mainly on rigid-body dynamics, with deformation and articulated contact receiving limited attention.Source: §5.5 · pp. 25–26

5.6 Physics-Guided Diffusion-based Generation

  • Physics guidance steers diffusion models toward physically feasible solutions during sampling, without changing their architectures or retraining their base models.The survey reports improved task performance while retaining the flexibility and generative capabilities of diffusion policies.Source: §5.6 · p. 26

5.7 Physics-Guided Neural Operators

  • Physics-guided neural operators supply physical structure through states and observables while leaving operator architectures and training losses unchanged.A differential-drive example shows purely data-driven Koopman models failing to recover nonholonomic geometry; the methods still require known kinematics and manually designed, platform-specific observables.Source: §5.7 · p. 26

6 Software Tools

The surveyed software ecosystem offers complementary tools for different physics-embedding routes and robotics tasks, rather than a unified workflow. Practitioners must combine frameworks, and several tools are oriented toward scientific computing or simulation rather than robotics deployment and closed-loop use.

6.1 Software for Model-Structured and Hybrid Physics-Learning Architectures

  • Model-structured and hybrid frameworks support physics-encoded architectures for dynamics learning, control, and estimation, with NeuroMANCER offering reusable building blocks and interconnected modules.NeuroMANCER supports workflows from model definition and dataset construction through training, evaluation, and deployment, including ONNX export.Source: §6.1 · p. 27, §6.1.2 · p. 28, §6.1.1 · p. 28

6.2 Software for Neural Differential Equations

  • Neural differential-equation tools support differentiable continuous-time models, but their abstractions range from low-level solver components to general scientific-modeling workflows.The surveyed tools include adjoint-based training and GPU support, while some lack reusable physical-system components or control-oriented design abstractions.Source: §6.2.1 · p. 28, §6.2.2 · p. 28, §6.2.3 · p. 28

6.3 Software for Equation Discovery & System Identification

  • Equation-discovery and system-identification software spans sparse-regression discovery, deep state-space identification, and classical polynomial NARMAX/NARX modeling.The surveyed tools include interpretable equation discovery, benchmarking workflows for nonlinear state-space models, and established structure-selection and parameter-estimation methods.Source: §6.3.1 · p. 28, §6.3.2 · p. 28, §6.3.3 · p. 28

6.4 Software for Physics-Informed Machine Learning

  • Physics-informed learning libraries automate differential-equation residuals or loss construction, but several focus on scientific computing or PDE surrogates rather than robotics workflows.The survey also notes limitations in control-oriented workflows and native deployment support for some libraries.Source: §6.4 · p. 28, §6.4.1 · p. 28, §6.4.2 · p. 28

6.5 Differentiable Simulation Engines

  • Differentiable simulation engines enable gradient-based learning and optimization through simulation, with different support for parallel rigid-body simulation, articulated contact, and robotics applications.The surveyed tools include accelerator-oriented parallel simulation, robotics-oriented hard-contact support, and lightweight C++/CUDA engines.Source: §6.5 · p. 29, §6.5.1 · p. 29, §6.5.2 · p. 29

6.6 Choosing a Software Framework

  • Framework choice depends on the embedding route and application: physics-informed tools are better supported, while inference-time physics guidance lacks dedicated software and is usually task-specific.The survey also reports no dedicated ready-to-use framework for DeLaN, LNNs, or HNNs; users commonly build them with general-purpose automatic differentiation libraries.Source: §6.6 · p. 29

6.7 Discussion

  • No existing framework supports physics-guided, physics-encoded, and physics-informed learning in one workflow spanning model definition, training, evaluation, deployment, and embedded execution.The survey describes a fragmented ecosystem and calls for interoperable abstractions, differentiable solvers, uncertainty quantification, and standardized benchmarking.Source: §6.7 · p. 29, §6.6 · p. 29

7 Open Research Questions and Challenges

The survey identifies open challenges in selecting and reusing physics priors, handling model mismatch, establishing interpretability and trust, evaluating physical consistency, and connecting learned models to safety guarantees. Existing findings suggest that suitable priors depend on the system and task, while standardized evaluation and formal assurance remain limited.

Challenge 1 – Generality versus Physics-based Inductive Bias

  • Strong structural priors can improve sample efficiency and plausible extrapolation with scarce data, while hybrid models offer a compromise; contact-rich, nonlinear systems require more flexibility than smooth rigid-body dynamics.The survey concludes that appropriate prior strength and fidelity are problem-dependent, but principled selection guidelines remain absent.Source: §7 · p. 30

Challenge 2 – Reusable and Scalable Physics Priors

  • Reusable and modular priors show promise across operating conditions and related robots, but transfer across different embodiments, contacts, sensing, and actuation still requires substantial redesign.Separating invariant physical structure from robot-specific components remains a central challenge.Source: §7 · p. 30

Challenge 3 – Physics Priors under Model Mismatch and Sim-to-Real Transfer

  • Embedded structure can facilitate transfer when underlying physics remains approximately invariant, but omitted dominant effects can create systematic errors that residual learning only partially compensates.Such mismatch can reduce parameter interpretability and weaken theoretical guarantees; failures may reveal assumptions needing refinement.Source: §7 · p. 30

Challenge 4 – Interpretability, Explainability, and Trust

  • Combining analytical models with learning can weaken the physical meaning of learned parameters, while explicit explainability and standardized quantitative interpretability and trust metrics remain limited.Many works infer explainability from structured architectures rather than testing it with attribution, counterfactual analyses, or user studies.Source: §7 · pp. 30–31

Challenge 5 – Assessing Physical Consistency

  • Physical-consistency metrics assess different properties, horizons, perturbations, and tasks, while evaluations remain tied to paper-specific datasets and setups.The survey calls for standardized metrics and benchmark suites covering conservation, constraints, long-horizon prediction, uncertainty calibration, and closed-loop behavior.Source: §7 · p. 31

Challenge 6 – Bridging Physics and Safety

  • Physical consistency alone does not establish safety, and only a few surveyed papers provide explicit theoretical guarantees; formal verification and safety certification remain largely unexplored.Explicit probabilistic treatment of model or prediction uncertainty is also comparatively uncommon.Source: §7 · p. 31

8 Future Research Directions

The survey identifies seven promising research directions for embedding physics in robot learning. They address combining prior routes, expanding scope and scale, and improving methods and supporting tools. [p10]

Direction 1 – Combining Multiple Physics-Prior Routes

  • Only 4% of reviewed works combine two physics-prior routes, and existing combinations are largely ad hoc; principled selection should reflect task, data, and prior system knowledge.Open questions include balancing hard architectural constraints against soft loss regularization and detecting redundant or conflicting priors.Source: §8 · pp. 31–32

Direction 2 – Physics-Aware Foundation Models

  • Most surveyed methods are trained or fine-tuned for one robot or application, motivating physics-aware foundation models that could generalize across tasks, environments, and embodiments.Existing approaches remain largely task- and robot-specific, and recent evidence suggests foundation models do not automatically acquire general physics understanding from data alone.Source: §8 · p. 32

Direction 3 – Differentiable Physics-Embedded Learning Pipelines

  • End-to-end differentiability can propagate task-level gradients across perception, prediction, planning, control, and simulation, but gradient fidelity remains challenging around contact and discontinuities.The survey also calls for online lifelong adaptation of priors during operation while preserving physical consistency and safety.Source: §8 · p. 32

Direction 4 – Physics-Embedded Learning for Contact-Rich Tasks

  • Contact-rich methods remain largely task-specific despite the need to represent impacts, friction, compliance, deformation, and hybrid mode transitions across robotic tasks.The survey calls for robust contact representations from partial observations and generalization across contact conditions, object properties, and robot morphologies.Source: §8 · p. 32

Direction 5 – Automated Discovery of Physics-Encoded Architectures

  • Automated discovery of physics-encoded architectures could reduce manual redesign by composing physically admissible building blocks, with users specifying candidates and learning finding their interconnections.Current architecture design typically requires engineering effort and domain expertise, and models are often tailored to a particular robot or application.Source: §8 · p. 32

Direction 6 – Scaling to High-DoF Systems

  • Scalability remains underexplored: 16 papers address legged robots, while 62% of reviewed papers concern manipulators and vehicles, typically modeled with fewer than ten DoFs.Future evaluation should include computational cost, long-horizon stability, constraint satisfaction, and transfer across morphologies and operating conditions.Source: §8 · p. 33
  • Scaling models should preserve computational efficiency and physical consistency by exploiting modularity, sparsity, equivariance, and reduced-order representations.Evaluation should also account for contact, compliance, underactuation, and unmodeled dynamics.Source: §8 · p. 33

Direction 7 – Open-Source Software Frameworks

  • Interoperable, modular software frameworks and standardized benchmarks are needed because current tools specialize in single paradigms, complicating combined workflows and reproducible comparisons.Suggested abstractions include physics and learning modules, differentiable solvers, uncertainty quantification, benchmarking, and deployment.Source: §8 · p. 33

9 Conclusions

The survey organizes physics-embedded robot learning into three complementary routes and finds that physics-encoded architectures account for the largest share of reviewed methods. It concludes that physics priors should complement data-driven learning, while substantial challenges remain. [p37] [p40]

9 Conclusions

  • Physics-encoded architectures represent the largest body of reviewed methods (71%), compared with physics-guided (16%) and physics-informed (13%); more than half were published between 2023 and 2025.Source: §9 · p. 33
  • The survey presents a unified taxonomy as a common conceptual foundation for a field fragmented across terminology, methods, software tools, and application domains.It envisions combining physics-guided, physics-encoded, and physics-informed components alongside physics-aware foundation models, scalable architectures, and interoperable software.Source: §9 · p. 33
  • Physics priors are robotics-specific inductive biases that complement, rather than replace, data-driven learning, with potential benefits for generalization, physical consistency, interpretability, and sample efficiency.No universally optimal embedding strategy has emerged; selection depends on prior fidelity, available data, and required closed-loop properties.Source: §9 · p. 33
  • The review identifies persistent challenges in standardized evaluation, real-world safety guarantees, transfer across robots and tasks, and broader platform coverage.The reviewed methods target manipulators or vehicles in 62% of cases, while only 7% address legged robots; just 4% combine multiple embedding routes, often ad hoc.Source: §9 · p. 33

Funding

The funding statement names support from the Alexander von Humboldt Foundation and the Italian FIS 2 Call. [p43] [p45]

Funding

  • The Alexander von Humboldt Foundation supported the work through a Humboldt Postdoctoral Fellowship.Source: Funding · p. 33
  • The work also received support through the Italian FIS 2 Call under Grant Assignment Decree No. 1236.The decree was adopted by the Italian Ministry of University and Research.Source: Funding · p. 34

10 List of Acronyms

The section expands acronyms for artificial intelligence, learning models, dynamical methods, and robotics tools. [p46] [p47] [p64] [p91]

List of Acronyms

  • AI means Artificial Intelligence; ANFIS means Adaptive Neuro-Fuzzy Inference System; BNN means Bayesian Neural Network; and CNN means Convolutional Neural Network.Source: §10 · p. 34
  • EQLN means Equation Learner Network; GPR means Gaussian Process Regression; SINDy means Sparse Identification of Nonlinear Dynam. Systems; and SysID means System Identification.Source: §10 · p. 34
  • MPC means Model Predictive Control; RL means Reinforcement Learning; VLA means Vision-Language-Action Model; and VWM means Video World Model.Source: §10 · p. 34
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