Constrained Regression Projection for Mass and Energy Balance
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Solution Overview
Problem
Existing regression models fail to preserve physical and chemical constraints in chemical and industrial processes, leading to unfeasible predictions such as mass increase or sudden temperature fluctuations, due to errors in measurement or numerical computational errors.
Innovation Solution
The approach involves constructing a constrained regression model by multiplying the solution of an unconstrained linear model by a matrix that enforces equality and gain constraints, ensuring that the model predictions respect mass balances, atom balances, and energy balances, using techniques like Lagrange multipliers and numerical optimization.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If prior art data driven machine learning models are used, then model flexibility and adaptability are improved, but physical and chemical constraints are violated leading to unfeasible predictions
Solution Approach 1:
The patent introduces a projection matrix as an intermediary component that maps unconstrained model predictions to a feasible space that satisfies physical and chemical constraints. This projection matrix acts as a mediator between the flexible machine learning model and the constraint requirements, allowing the model to maintain its adaptability while ensuring reliable predictions that obey mass balance, atom balance, and energy balance equations.
Solution Approach 2:
The patent transforms the prediction problem by changing the parameter space through projection. Instead of directly predicting constrained values, the model predicts in an unconstrained space and then applies a projection transformation to obtain feasible predictions. This parameter transformation allows the model to maintain flexibility in learning while guaranteeing constraint satisfaction in the output.
2Reliability
If numerical optimization methods are used to incorporate constraints, then constraint satisfaction is improved, but computational complexity and solution existence are worsened
Solution Approach 1:
The patent extracts the constraint satisfaction problem from the optimization process by using a projection matrix that directly enforces constraints. Instead of using complex numerical optimization methods that require solving large systems of equations, the approach extracts constraint requirements and embeds them into a projection operation that can be applied directly to model predictions, significantly reducing computational complexity.
Solution Approach 2:
The patent replaces the mechanical numerical optimization system with a mathematical projection operation. Rather than iteratively solving optimization problems with complex constraints, the method substitutes a direct projection matrix multiplication that efficiently maps predictions to the feasible space, eliminating the need for complex optimization algorithms.
3Reliability
If penalty terms are added to loss function, then constraint violation is reduced, but exact constraint satisfaction and model accuracy are worsened
Solution Approach 1:
Instead of trying to prevent constraint violations through penalty terms in the loss function, the patent inverts the approach by first allowing unconstrained predictions and then projecting them onto the feasible space. This inversion ensures exact constraint satisfaction while maintaining model accuracy, as the projection operation preserves the underlying prediction quality while enforcing constraints.
Data Source
AI summary
Computer implemented methods and systems incorporate physics-based and/or chemistry-based constraints into a model of a chemical, physical, or industrial process. The model is derived from a representative dataset of the subject process. The constrained model provides predictions of process behavior that are guaranteed to be consistent with incorporated constraints such as mass balances, atom balances, and/or energy balances while being less computationally intensive than equivalent first principle models. The constrained model can be constructed by matrix multiplication, namely multiplying the solution of an unconstrained linear model by a matrix that enforces the constraints. Improved process control models result, as well as improved process modeling and simulation models result.