Hybrid modelling with physics and machine learning
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What we can help you with
Physical models describe known relationships, but they can be computationally expensive or lack an accurate description of particular processes. Data can help calibrate, supplement or simplify these models. We develop hybrid approaches that combine numerical simulation and machine learning according to what is known about the system, the data available and the intended use of the model.
We can help you:
- Develop fast, physics-based models trained on measurements or detailed simulations.
- Replace expensive computations with surrogate models for analysis and optimisation.
- Learn unknown relationships or correct systematic model discrepancies using data.
- Integrate neural networks and other learned components into existing simulators.
- Assess model accuracy, physical properties and performance under new conditions.
Do you have a simulator that is too expensive to run, or data you would like to make better use of in a physical model? Contact our Applied Computational Science research group.
Combining physical structure with learning from data
Hybrid modelling takes several forms. A physical simulator can itself become a trainable model, a neural network can represent an unknown part of the model, or a surrogate can learn to reproduce selected outputs from a more detailed simulation. The choice depends on the available data, computational cost and the physical properties that need to be preserved.
Our approach builds on in-depth knowledge of numerical methods and simulator development. We investigate how physical structure can be incorporated into models and their training, and how differentiable simulators and sensitivity analysis can support efficient calibration of both physical and learned components.
A substantial part of our work concerns subsurface flow, including reservoir simulation, CO₂ storage and geothermal energy. We also work with other dynamic systems and flow problems. The examples below illustrate different ways of combining models and data.
What do we do?
We develop hybrid modeling methods that integrate physical insight with machine learning to create fast, accurate flow models for complex systems. Our primary focus is on subsurface applications—including hydrocarbon recovery, CO₂ storage, geothermal energy, and gas storage—where uncertainty is high and data availability is limited.
🧠 Treating the simulator as a neural network |
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In our CGNet approach, we use a fully differentiable flow simulator as a trainable model—conceptually analogous to a neural network. The model architecture is defined by the computational graph induced by a finite-volume discretization over a coarse grid (or a coarse partition of a more accurate grid), where physical properties such as pore volumes and transmissibilities act as tunable weights. These are calibrated using gradient-based optimization, with automatic differentiation and adjoint methods providing the equivalent of backpropagation. CGNet can be trained to match observed field data or to emulate high-fidelity simulation output—offering rapid, physics-consistent predictions at reduced computational cost. |
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🔁 Other hybrid flow models |
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CGNet is one of several hybrid approaches we develop. We also work with numerical interwell network models like GPSNet, FlowNet, and StellNet, which use simplified flow networks trained on field or simulation data to predict reservoir behavior quickly. For unconventional resources such as shale oil, we use calibrated 1D models to predict production from hydraulically fractured, ultra-low-permeability systems, capturing fracture–matrix interactions and transient flow dynamics. |
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📚 Standard machine learning approaches |
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We have experience applying established techniques like Physics-Informed Neural Networks (PINNs), Fourier Neural Operators (FNOs), Pseudo-Hamiltonian Neural Networks (PHNN), and other physics-aware architectures. These methods can be valuable when parts of the physics are unknown or when full-scale simulation is too computationally expensive. In addition, we have developed novel and fast surrogate models for estimating observables from simulators by combining machine learning with state-of-the-art uncertainty quantification techniques. These surrogate models enable efficient gradient-based optimization even on black-box simulators. By utilizing multi-fidelity data for training, we achieve better accuracy for the same computational cost. |
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🧩 System identification and inverse modeling |
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We explore how to learn governing equations directly from data, including system identification for non-linear PDEs. Our preliminary tests enforce known physical constraints by altering assumptions on the learned Hamiltonian, yielding interpretable and physically plausible models. We are also developing methods to embed neural networks implicitly into simulators, enabling solutions to inverse problems by combining differentiable simulators, functional approximations (e.g., neural networks), and a priori constraints. |
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🔄 Neural networks as correctors |
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We are currently exploring predictor–corrector hybrid frameworks, where a calibrated simulator or a physics-based model like CGNet serves as the predictor and a neural network acts as the corrector. The predictor enforces core physical principles, while the corrector improves accuracy by learning from mismatches with observed data. How the correction is formulated determines which physical properties are preserved. We investigate how improved agreement with data can be combined with appropriate physical constraints. |
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🌍Supporting physics-informed digital twins |
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Our hybrid models can provide a foundation for digital twins of subsurface systems—dynamic, real-time simulations that integrate operational data and sensor inputs. These digital twins are designed to improve understanding of complex processes and support more effective decision-making throughout the asset lifecycle. By combining fast surrogate models with continuous data updates, they enable forecasting, monitoring, and optimization under uncertainty. Much of our work in this area has focused on geothermal energy systems, where real-time modeling and data integration are essential for efficient resource management, performance prediction, and operational planning. |
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