Draft:Physics-Enhanced Machine Learning

Physics-Enhanced Machine Learning

Physics-Enhanced Machine Learning (PEML) is a subfield of machine learning that combines physics-based modelling and data-driven algorithms, and aims to combine the complimentary strengths of physical knowledge and machine learning algorithms to overcome the limitations of using either approach alone. PEML incorporates known physical laws and domain knowledge into the machine learning process so that models not only fit observed data patterns, but also respect the governing physics equations of the model.

This concept is closely related to scientific machine learning (SciML), Physics-Informed Machine Learning (PIML), Physics-Enhanced Artificial Intelligence (PEAI) and Physics-Guided Machine Learning, but is distinct in its emphasis on enhancing predictive capability through the use of physics-based components in hybrid or constrained models.

Background and Motivation

The motivation for PEML arose from challenges often encountered in engineering and real-world problems: The volume of useful data is generally limited; predictions obtained from modelling can make accurate, but physically implausible predictions that violate known physics; uncertainties could not easily be dealt with; and traditional machine learning models are not very explainable or interpretable. By infusing physics into machine learning, it ensures better generalisation to unseen conditions, improved physical consistency of predictions, and a greater explainability of the learnt models. This approach has gained traction since 2019, where techniques have been used to encompass strategies where "prior physics knowledge is embedded to the learner" , particularly in dealing with dynamical systems in engineering.

Early examples of the need for PEML arose from fields such as structural mechanics and environmental science, where purely data-driven methods struggled with limited data or lacked reliability. For instance, in structural engineering, traditional physics-based simulations can be very accurate but often require costly modelling and still face uncertainty in loads or material properties; on the other hand, data-driven models may fit experimental data but fail to generalise outside those conditions. PEML approaches were developed to bridge this gap, effectively creating a "spectrum" between the extremes of purely physics-based (white-box modelling) and purely data-driven (black-box modelling), known as grey-box or hybrid modelling. In practice, this means a PEML model can leverage governing equations or simulation data to inform the learning process, thus requiring less training data and yielding outputs that obey physical laws.

Methods and Techniques

PEML encompasses a range of methods that integrate domain-specific physical knowledge into the machine learning process. These techniques differ in how physics is incorporated, whether through loss functions, model structures, feature design, or data generation, and can be summarised into three different categories: Physics-Informed, Physics-Guided, and Physics-Encoded Machine Learning.

Physics-Informed Learning

See also: Physics-informed neural networks

Physics-Informed learning techniques integrate physical laws directly into the machine learning process to simulate complex systems using partial differential equations (PDEs) and embedding physical constraints directly into a machine learning algorithm, such as a neural network. An example of physics-informed learning is through the use of Physics-Informed Neural Networks (PINNs), which implement composite loss functions that balance errors with PDE residuals, effectively blending sparse observations with physical constraints. They are most suitable for irregular geometries due to their ability to operate in a meshless paradigm by sampling random collocation points. Physics-informed learning excels in multi-physics scenarios such as electroconvection, molecular dynamics, and real-time 4D flow reconstruction from MRI observations.

Physics-Guided Learning

In many PEML approaches, physical knowledge is introduced into the learning process not by altering the model itself, but through data preprocessing and feature engineering. This strategy enables conventional machine learning algorithms to work with inputs that already encode important physical structure, enhancing both accuracy and interpretability. Common techniques include:

1. Physics-based feature extraction. Raw data is transformed into features with physical meaning, such as dimensionless numbers (e.g. Reynolds number or Mach number), wavelet coefficients, or energy spectra. For example, Mohan et al. used a wavelet transform to extract turbulence-related features from velocity fields, embedding known physics of turbulent cascades into the model inputs.
2. Simulation or theory-driven feature augmentation. Outputs of simplified physical models (or residuals between observed and predicted behaviour) are used as additional features, reducing the learning burden of the machine learning model by letting it focus only on the discrepancy. This technique has been used in chemical kinetics applications, where delta learning was applied to graph neural networks (GNNs) to enhance activation energy predictions in chemical reactions, or in Quantitative Structure-Activity Relationships (QSARs), where molecular descriptors derived from quantum chemistry calculations or physical models are used as features to predict chemical properties.
3. Physical domain transformations. Data is mapped into domains where physics-relevant patterns are more easily captured. For example, signal processing often employs Fourier transforms to reveal frequency content, allowing oscillatory features to be revealed. This enables machine learning algorithms, such as convolutional neural networks (CNNs) to apply standard vision models, yielding better generalisation and efficiency by learning from spectrograms instead of raw waveforms.

These preprocessing methods are especially useful when physical insight is available, but the system is too complex for fully mechanistic modelling. By encoding physics into the data, standard machine learning architectures such as multi-layer perceptrons (MLPs) can be trained without needing architecture-specific changes. This class of approach enables physics-guided learning, where training data already obeys physical laws. As a result, the learned mapping is inherently constrained by the input features, and the model does not need to discover fundamental physical relationships from scratch, since key patterns are already embedded in the data.

Physics-Encoded Learning

Physics-encoded learning, otherwise known as hybrid modelling, combines physics-based components with data-driven components in a singular framework. This approach is useful when the underlying physical laws are partially understood but insufficient to describe the full system behaviour, and are computationally expensive to simulate. In such methods, the final model $M$ integrates the physics-based model $G$ and the data-driven correction term $D$, along with additional biases to narrow the solution space to only contain physically plausible outputs such that the system is in the form:$$f(x)\approx M[G(\mathbf {x} _{model},\mathbf {\theta } _{p})\cup D(\mathbf {x} _{measured},\mathbf {y} _{measured},\mathbf {\theta } _{D})+biases]$$Common examples of physics-encoded learning include Gaussian Process (GP) latent force models and Physics-Informed Sparse Identification of Nonlinear Dynamics (PhI-SINDy), which have been used to model multiple degree-of-freedom (MDOF) oscillators with multiple Coulomb friction contacts under harmonic load using both synthetic and experimental noisy experiments with multiple sources of discontinuous nonlinearities.

Applications of Physics-Enhanced Machine Learning

PEML methods have moved beyond theoretical development and are now actively deployed in real-world systems across engineering, biology, chemistry, physics, scientific discovery, and computer science, to name a few applications. These applications are especially valuable in high-stakes or data-scarce environments where traditional machine learning or purely physics-based models may fall short.

Wind Turbine Structural Monitoring

PEML has been applied to predict fatigue loads in wind turbine blades under wake steering control (WSC), a strategy that improves wind farms efficiency by intentionally misaligning turbine yaw angles to reduce wake interference. While WSC can enhance power output, it also introduces additional fatigue loads on downstream turbines, complicating structural health monitoring. Traditional methods, such as look-up-tables (LUTs), often fail to capture the nonlinear dynamics of wake-induced loading. A recent approach addressed this by using Gaussian process (GP) models trained on physics-informed features, including damage-equivalent loads (DELs) derived from Rainflow Counting and the Palmgren-Miner rule. These GPs provided probabilistic fatigue predictions with improved accuracy. Compared to LUTs, the PEML model reduced the root mean square error (RMSE) by 13.99% for edgewise moments and by 51.87% for flapwise moments, highlighting the value of incorporating fatigue physics into machine learning-based predictive maintenance.

Tuned Mass Damper Optimisation

Tuned Mass Dampers (TMDs) are widely used to mitigate structural vibrations in tall buildings during seismic events. Traditional physics-based design methods, such as the Den Hartog approach, assume linear structural behaviour and do not fully capture the effects of nonlinear dynamics or variable seismic loads. Conversely, purely data-driven optimisation techniques may lack physical constraints, resulting in unrealistic or inefficient damping configurations. To address this, researchers developed a PEML framework based on a generative adversarial network (GAN) architecture. The system incorporates a physical evaluation network into the GAN loop to guide the generation of TMD parameters (the natural frequency and damping ratio) under realistic seismic excitations. This approach was tested on both linear shear-type structures and nonlinear moment-resisting frames. Compared to traditional Particle Swarm Optimisation (PSO), the physics-enhanced GAN achieved a 24.14% reduction in displacement under seismic loading while reducing computational cost by 80%, demonstrating the effectiveness of hybrid machine learning approaches in structural vibration control.

Satellite Attitude Control

In spacecraft missions involving dynamic payload changes, such as active debris removal, traditional attitude control systems (ACS) that rely on fixed mass and inertia properties may struggle to maintain stability. A study published in Frontiers in Robotics and AI proposed a PEML approach using deep reinforcement learning (DRL) to address this challenge. The method integrated physics-based simulation with DRL algorithms such as Proximal Policy Optimization (PPO) and Soft Actor-Critic (SAC), and was trained using the Basilisk high-fidelity spacecraft simulator, which models Newtonian rotational dynamics and reaction wheel behaviour. The approach incorporated "stacked observations," feeding sequences of sensor readings (e.g. angular velocities and torques) into the learning model to enable inference of unknown mass properties over time. Compared to conventional proportional–integral–derivative (PID) controllers, DRL controllers with stacked observations achieved improved control performance, particularly in scenarios involving unknown or varying mass distributions. In simulation, the SAC controller with stacking reduced attitude error by up to 78° and settled the spacecraft 26 seconds faster than the PID controller. These results highlight the potential of PEML methods for improving control robustness under uncertain spacecraft dynamics.

Aircraft Flight Path Optimisation

Accurate upper-air wind field prediction is essential for optimising aircraft trajectories to reduce fuel consumption and flight time. Traditional numerical weather prediction (NWP) methods, while physically rigorous, are computationally expensive and limited in short-term forecasting. A recent study proposed a method that integrates a Predictive Recurrent Neural Network (PredRNN) with an improved A* pathfinding algorithm to generate efficient flight routes in dynamic wind conditions. PredRNN was trained on ERA5 wind data at cruising altitudes of 5,500m along major Chinese airline routes using a loss function informed by Navier-Stokes equations. The resulting wind field forecasts enables the A* algorithm to avoid zones of high turbulence and optimise routes in real-time for up to 10 hours in advance. Compared to standard neural network and physics-based approaches, this framework improved forecasting accuracy and produced safer, more fuel-efficient trajectories.

Streamflow Discharge Forecasting

Accurate river discharge forecasting is critical for flood mitigation, as well as waterway management and infrastructure planning. Physics-based hydrological models such as RAPID (Routing Application for Parallel Computation of Discharge) simulate river flow based on the Muskingum algorithm but often make assumptions such as linear process modelling and reliance on adjacent inflows, simplifying the problem. These limitations can lead to deviations from observed discharge values, especially in complex or ungauged river networks. To address this, a PEML approach was proposed that integrates RAPID with data-driven models using delta learning and data augmentation techniques. These hybrid models combine physical runoff simulations with machine learning algorithms, including Gaussian Process Nonlinear Autoregressive with Exogenous Inputs (GP-NARX), neural networks, and bidirectional LSTMs. The goal is to compensate for uncertainties in the RAPID model by learning discrepancies between predicted and gauged discharge values and using additional basin-wide runoff data to inform forecasts. The study demonstrated that the hybrid PEML models significantly outperformed RAPID alone, improving discharge prediction accuracy by a factor of four to seven across various river systems in the United States. By leveraging both physical principles and basin-wide hydrological data, the approach enables robust, long-range forecasting in data-limited conditions and enhances the reliability of streamflow predictions for gauged rivers.

CO₂ Flooding Recovery Prediction

PEML has been applied to predict oil recovery during immiscible CO₂ flooding in sandstone reservoirs, a widely used enhanced oil recovery (EOR) method. Traditional core-flooding experiments and physics-based models, while informative, often rely on simplifying assumptions regarding flow dynamics which can limit their predictive accuracy. To improve generalisation, researchers developed a PEML framework combining experimental data with physically informed features expressed through dimensionless numbers, which include the capillary number, relative radius (based on porosity and permeability), injection pressure ratio, and oil composition number. The model was trained on core-flooding datasets spanning a wide range of reservoir conditions: porosity (10.8-37.2%), permeability (1-18,000 mD), injection pressures (2.73-11.44 MPa), flow rates, and various crude oil types. Rather than relying on individual parameters, the PEML model used a grouped dimensionless formulation to represent the combined physical forces governing displacement efficiency. A logarithmic correlation was found between these grouped parameters and the oil recovery factor, achieving strong agreement with experimental results (81% confidence). This approach demonstrated improved accuracy over traditional methods and highlighted the benefits of embedding domain knowledge into machine learning for more robust EOR performance prediction.

Illumination Harmonisation and Editing

In image processing and computer vision, PEML has been used to improve illumination harmonisation and editing tasks. Traditional graphics models are often computationally expensive and may struggle to generalise to diverse real-world lighting conditions. Conversely, standard diffusion-based models are powerful for generative tasks, but can alter intrinsic image properties such as albedo or reflectance, leading to unrealistic visual artifacts. To address these limitations, researchers proposed a PEML-based training strategy known as Imposing Consistent Light (IC-Light) transport. This method incorporates physical light transport theory into the training of diffusion-based illumination models by enforcing a consistency principle: the linear blending of different lighting conditions should reflect physically plausible results. By embedding this constraint during training, the model learns to modify illumination without distorting other visual features of the image. IC-Light was applied to a large-scale training regime involving over 10 million samples, including real photographs, rendered data, and in-the-wild synthetic augmentations. The model was benchmarked against several baselines (e.g. SwitchLight and DiLightNet), achieving state-of-the-art results in perceptual quality (LPIPS = 0.1025), while maintaining balanced performance in PSNR (23.72) and SSIM (0.8513). This PEML approach enables more stable and scalable illumination editing while being physically consistent, supporting applications in content creation and digital design.

Challenges and Limitations

Despite its advantages, PEML faces several challenges that limit its scalability and general adoption.

One major issue is the lack of standardised benchmarks for evaluating PEML models.