Declarative Differential Models (DDM)
Declarative Differential Models (DDM) refer to a modeling approach where the entire system's behavior—states, transitions, constraints, and objectives—is described declaratively and encoded directly within a set of differential equations or differential-algebraic equations (DAEs).
Key Features of DDM:
Declarative Nature:
- The system is described by its rules and relationships, not step-by-step instructions.
- These rules are translated into continuous or discrete dynamics that govern the system's evolution.
Constraint Embedding:
- Logical and physical constraints (e.g., conservation laws, capacity limits) are directly included in the model equations.
- This ensures the system operates within valid bounds without requiring external enforcement.
Dynamic and Adaptive Behavior:
- DDMs can handle both deterministic and probabilistic transitions, making them suitable for hybrid systems (e.g., mixing continuous flows and discrete events).
Optimization-Friendly:
- DDMs naturally integrate with optimization techniques, allowing parameters (e.g., transition rates) to be learned or fine-tuned directly within the model.
Examples and Applications:
- Petri Nets: Encoding resource flows, transitions, and state changes as ODEs.
- Hybrid Systems: Combining discrete decision-making with continuous dynamics.
- Physics-Inspired Machine Learning: Embedding physical constraints within Neural ODEs.
- Optimal Control: Modeling decision problems directly in the dynamics for optimization.
Why Declarative?
The term “declarative” emphasizes that the model focuses on what the system does (rules and behaviors) rather than how it does it (procedural steps).
Use Case Example:
In a knapsack optimization problem:
- States represent item availability, weight, and value.
- Transitions represent decisions to include/exclude items.
- Constraints (e.g., weight capacity) are encoded directly into the model dynamics, making the system solvable and interpretable.
See the python notebook