Evolutional Deep Neural Networks for Long-Time PDE Prediction
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Solution Overview
Problem
Current methods for predicting dynamics over long times using neural networks, such as PINNs, face challenges with computational complexity and optimization issues, especially for long-time multiscale problems like chaotic turbulent flows, due to the need for extensive training and high storage requirements.
Innovation Solution
The Evolutional Deep Neural Network (EDNN) approach, which represents the solution in space at a single instant and evolves the neural network parameters using governing equations to predict future states without further training, allowing for indefinite long-time predictions while enforcing boundary conditions as hard constraints.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Duration of action of moving object
If physics-informed neural networks (PINNs) are used to represent spatio-temporal solutions over long time horizons, then the network can capture temporal evolution, but the neural network complexity and optimization space dimension grow prohibitively large
Solution Approach 1:
The patent segments the spatio-temporal solution representation by separating spatial and temporal dimensions. A neural network represents the spatial solution at a single time instant, while temporal evolution is achieved by evolving the network parameters according to the governing PDEs. This segmentation prevents the exponential growth of network complexity that would occur if the entire spatio-temporal domain were represented by a single network.
Solution Approach 2:
The patent transitions from representing the solution in the original spatial dimensions to representing it in the neural network parameter space. By evolving parameters rather than expanding the network architecture over time, the method effectively adds a temporal dimension to the parameter evolution process without increasing spatial complexity.
2Duration of action of moving object
If physics-informed neural networks are trained to minimize PDE residuals over extended time horizons, then temporal evolution can be captured, but computational cost and storage requirements become prohibitive
Solution Approach 1:
The patent performs preliminary training of the neural network to represent the initial spatial condition. Once trained, the network parameters are evolved forward in time using the governing PDEs without requiring additional training computations. This preliminary action approach eliminates the need for continuous expensive optimization over extended time horizons.
Solution Approach 2:
The patent replaces the iterative optimization mechanism (gradient descent for minimizing PDE residuals) with a direct parameter evolution approach based on the governing equations. Instead of mechanically iterating through optimization steps to capture temporal evolution, the method substitutes this with analytical or semi-an analytical parameter updates derived from the PDEs themselves.
3Productivity
If a single neural network represents the entire spatio-temporal solution, then the approach is computationally efficient for short times, but it cannot explicitly specify temporal evolution or causality
Solution Approach 1:
The patent introduces dynamics into the neural network by making the parameters time-dependent. Rather than a static network representing the entire spatio-temporal domain, the network parameters evolve dynamically according to the governing PDEs, explicitly capturing temporal evolution and causality while maintaining computational efficiency.
Data Source
AI summary
Some embodiments provide a method of predicting a state of a system that is represented by a partial differential equation. The method comprises training a neural network for an initial state of said system to obtain a set of neural network parameters to provide a spatial representation of said system at an initial time. The method further comprises modifying said parameters for intermediate times between said initial time and a prediction time such that each modified set of parameters is used to provide a respective spatial representation of said system at each corresponding intermediate time using said neural network. The method further comprises modifying said set of parameters to provide a prediction set of parameters that is used to provide a predicted spatial representation of said system at said prediction time using said neural network.