Neural Reduced-Order Modeling for Stable PDE System Control
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Solution Overview
Problem
Existing control methods for high-dimensional physical systems, such as HVAC and gas leakage detection systems, face challenges due to nonlinear dynamics and the need for large datasets, often failing to capture the physics of the system and requiring proprietary software access, leading to inaccurate and inefficient control.
Innovation Solution
A neural network model with an autoencoder architecture is trained using time series data and collocation points to generate a non-intrusive reduced order model (ROM) that captures the system's physics, allowing for efficient control without requiring access to proprietary software.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional control methods are used for high-dimensional physical systems, then control stability can be achieved, but the systems require large datasets and fail to capture the physics of the system
Solution Approach 1:
The patent transforms the control approach by changing the parameter representation from raw high-dimensional state data to physics-informed latent features. The neural network learns a mapping that transforms system states into a reduced-order latent space where physics principles are embedded as constraints, allowing the controller to capture essential physical behaviors while reducing data requirements.
Solution Approach 2:
The patent introduces a physics-informed neural network as an intermediary between the high-dimensional physical system and the controller. This intermediary model learns the underlying physics relationships from data while incorporating domain knowledge through physics-informed loss functions, enabling accurate control without requiring the controller to directly process large volumes of raw system data.
2Productivity
If data-driven control methods are used, then large quantities of data are required, but the models do not capture the physics of the system
Solution Approach 1:
The patent creates a composite control model that combines data-driven neural network components with physics-informed constraints. The model integrates learned representations from training data with explicit physics principles encoded as loss function terms, resulting in a hybrid approach that achieves both high productivity through efficient data utilization and accurate physics capture.
Solution Approach 2:
The patent implements physics-informed loss functions that provide continuous feedback during the training process. These loss functions incorporate physics constraints that guide the neural network learning toward solutions that satisfy physical laws, ensuring that the model captures genuine physics relationships rather than merely fitting training data patterns.
3Measurement precision
If nonlinear models are used to describe system dynamics, then accuracy is improved, but the models are difficult to design and use in real-time
Solution Approach 1:
The patent extracts the essential nonlinear dynamics from the complex high-dimensional system by projecting them into a reduced-order latent space. The neural network learns to identify and extract the dominant nonlinear behaviors that govern system dynamics, representing them in a simplified form that maintains accuracy while enabling real-time control applications.
Solution Approach 2:
The patent transforms the problem by changing dimensions - moving from high-dimensional state space to a lower-dimensional latent space where nonlinear dynamics can be more efficiently represented. This dimensional transformation allows the system to capture complex nonlinear behaviors with fewer parameters, reducing model design and computational complexity.
4Productivity
If reduced order modeling is applied, then computational efficiency is improved, but accuracy in capturing high-dimensional physics may be reduced
Solution Approach 1:
The patent changes the parameterization approach by learning optimal latent space representations that efficiently encode high-dimensional physics. Instead of using fixed reduced-order models, the neural network adapts the parameter transformation to preserve essential physics information while achieving computational efficiency through dimensionality reduction.
Solution Approach 2:
The patent applies local quality by ensuring that the reduced-order model captures physics accurately in local regions of the state space through physics-informed training. The model focuses computational resources on learning accurate representations in critical regions while maintaining overall efficiency, allowing high accuracy in capturing essential physics without requiring full high-dimensional modeling everywhere.
Data Source
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AI summary
A system and method are provided for training neural network for controlling operation of system having non-linear dynamics represented by partial differential equations (PDEs). The method comprises collecting digital representation of time series data indicative of instances of function space of the system and measurements of state of the operation of the system. Collocation points corresponding to solutions of the PDE are generated. The neural network is trained using training data including the collected time series data and the collocation points to train parameters of non-linear operator. The neural network has autoencoder architecture including encoder to encode each instance of the training data into latent space, the non-linear operator to propagate the encoded instances into the latent space with transformation determined by parameters of the non-linear operator, and decoder to decode the transformed encoded instances of the training data to minimize a hybrid loss function.