Reversible-Heun FBSDE Training for Stable, Memory-Efficient Generative Models
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Solution Overview
Problem
Existing deep FBSDE frameworks face memory complexity issues (O(N)) and gradient vanishing/exploding problems for long time horizons, and existing SDE solvers lose gradient information or require non-differentiable loss functions.
Innovation Solution
A Reversible-Heun SDE solver is used to solve SB-FBSDEs, allowing for O(1) memory complexity and accurate gradient computation through algebraic reversibility, combined with the stochastic adjoint method for end-to-end differentiability.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If discretize-then-optimize approach is used to solve FBSDEs with deep neural networks, then the method can solve systems of FBSDEs derived from HJB PDEs, but memory complexity increases to O(N) where N is the number of time steps, leading to Out-Of-Memory problems
Solution Approach 1:
The computational process is segmented into forward propagation and backward propagation phases. During forward propagation, trajectories are generated and intermediate states are stored. During backward propagation, gradients are computed using the Reversible-Heun solver which reconstructs past states algebraically without storing the entire computational graph. This segmentation allows solving FBSDEs while reducing memory complexity from O(N) to O(1).
Solution Approach 2:
The patent inverts the traditional approach by using algebraic reversibility to reconstruct past states from future states during backward propagation. Instead of storing all intermediate computational states from the forward pass, the Reversible-Heun solver algebraically inverts the forward dynamics to recover gradient information, thereby solving the memory complexity problem while maintaining the ability to solve FBSDEs.
2Duration of action of moving object
If discretize-then-optimize approach is used for long time horizon tasks, then the method can handle extended time periods, but gradient-vanishing and gradient-exploding problems occur
Solution Approach 1:
The Reversible-Heun solver implements a feedback mechanism where the backward propagation uses the final state and gradient information to algebraically reconstruct intermediate states. This feedback loop ensures that gradient information is preserved and propagated accurately through the entire time horizon, preventing gradient-vanishing and gradient-exploding problems while enabling long time horizon tasks.
3Quantity of substance
If memory buffer is used to store past trajectory data, then past trajectory information is preserved, but gradient information is lost due to discarding computational graphs
Solution Approach 1:
The Reversible-Heun solver acts as an intermediary that connects the stored trajectory data with the gradient computation. Instead of directly using stored trajectories which lose gradient information, the solver algebraically reconstructs the computational path, serving as a mediator that recovers gradient information from the stored states without requiring the original computational graphs.
4Quantity of substance
If algebraic reversibility of Reversible-Heun solver is used, then O(1) memory complexity is achieved, but the method must maintain exact algebraic reversibility properties
Solution Approach 1:
The Reversible-Heun solver changes the parameter representation by using algebraic relationships between states rather than storing raw trajectory data. By transforming the problem into an algebraic reconstruction framework where states are represented as functions of each other, the method achieves O(1) memory complexity while the algebraic structure itself manages the complexity of maintaining reversibility.
Data Source
AI summary
A clean example to be learned is sampled, the clean example being from an input set of training data to be used to train the generative model. Initial values are computed using the sampled clean example and the generative model. The initial values from the clean example and the computed initial values are fed to a Reversible-Heun (RH) Stochastic Differential Equation (SDE) solver to forward propagate using Schrodinger Bridge Forward-Backward Stochastic Differential Equations (SB-FBSDEs) computed for the generative model, producing predicted output values. A loss function is computed to compare the initial values and the predicted output values. A reverse of the reversible Heun SDE solver is used for, solving the stochastic adjoint SDE to compute the gradient and update the weights of the generative model.