Physics-Informed Neural Network Control for PDE System Dynamics
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Solution Overview
Problem
Existing control methods for high-dimensional physical systems with nonlinear dynamics struggle to accurately capture the physics of the systems, often requiring large datasets and resulting in suboptimal control policies that do not consider the underlying physics, leading to instability and inefficiency.
Innovation Solution
A neural network model with an autoencoder architecture, incorporating a Koopman operator-based linear predictor, is trained using time series data to linearize nonlinear dynamics, allowing for accurate representation and control of the systems, while reducing the need for large datasets and incorporating physical constraints.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If data-driven control methods are used to construct control policies directly from operational data, then the control policy can be obtained without an intermediate model-building step, but large quantities of data are required and the controller does not capture the physics of the system
Solution Approach 1:
The patent introduces a physics-informed neural network as an intermediary model that bridges operational data and control policies. This intermediate model incorporates physical laws and constraints to generate control policies without requiring large quantities of training data, thus resolving the contradiction between ease of control policy design and data quantity requirements
Solution Approach 2:
The patent changes the parameters of the neural network by incorporating physics-based constraints and differential equations into the network architecture and loss function. This transformation allows the model to learn from smaller datasets while ensuring physical consistency, thereby reducing the quantity of data required while maintaining ease of control policy design
2Reliability
If nonlinear models of system dynamics are used to accurately represent physical systems, then the model can capture the physics of the system, but the model becomes difficult to design and use in real-time control
Solution Approach 1:
The patent replaces complex nonlinear model design and real-time solution with a neural network-based approach. The neural network is trained offline to approximate the nonlinear dynamics, and during real-time control, only forward propagation is required. This substitution maintains accuracy of system representation while dramatically reducing real-time computational complexity
Solution Approach 2:
The patent performs the complex model design and training work in advance during an offline phase. The neural network is trained on historical data and physics constraints before deployment. During real-time operation, the pre-trained model provides accurate predictions without requiring complex real-time computations, thus resolving the contradiction between reliability and device complexity
3Ease of operation
If conventional control methods are used that map system state directly to control commands, then the control policy can be implemented simply, but the control policy does not consider the physics of the system leading to instability
Solution Approach 1:
The patent modifies the control policy parameters by incorporating physics-informed constraints into the neural network's loss function and architecture. The network learns to map states to commands while respecting physical laws, ensuring stability. This allows simple implementation through the neural network while maintaining physical consistency and system stability
Solution Approach 2:
The patent implements physics-based feedback mechanisms within the neural network training process. The loss function includes terms that penalize violations of physical constraints and promote stable behavior. This feedback during training ensures the resulting control policy maintains simplicity for implementation while guaranteeing stability through embedded physical principles
Data Source
AI summary
Embodiments of the present disclosure provide a method of training a neural network model for controlling an operation of a system represented by partial differential equations (PDEs). The method comprises collecting digital representation of time series data indicative of measurements of the operation of the system at different instances of time. The method further comprises training the neural network model having an autoencoder architecture including an encoder to encode the digital representation into a latent space, a linear predictor to propagate the digital representation into the latent space, and a decoder to decode the digital representation to minimize a loss function including a prediction error between outputs of the neural network model decoding measurements of the operation at an instant of time and measurements of the operation collected at a subsequent instance of time, and a residual factor of the PDE having eigenvalues dependent on parameters of the linear predictor.