Unified Symbolic-Topological Framework for Physical Systems

Abstract

Solving the fundamental challenges in physics and engineering requires not only robust data but also deeper, verifiable theoretical foundations. This project proposes a unified, formal framework that integrates advanced mathematics (algebraic topology, category theory), symbolic reasoning, and cutting-edge artificial intelligence (neural-symbolic AI) to describe, simulate, and reason about physical systems. The aim is to move towards a framework for verifiable scientific discovery, enabling the automatic generation and proof of physical laws and system behaviors.

Project Vision & Motivation

Traditional scientific computing often relies on numerical simulations that lack formal verification or transparent reasoning. This framework addresses this by building a bridge between symbolic representations of physical laws, rigorous mathematical structures, and powerful AI methods. I envision a future where complex physical systems can be designed, analyzed, and controlled with formal guarantees.

Key Features (Current & Planned)

Phase 1: Formalization of Physical Abstraction (In Progress)

• Domain-Specific Language (DSL): A LaTeX-style syntax for defining physical entities (particles, fields), their properties (units, types), and laws (PDEs, symmetries).
• Categorical Intermediate Representation (IR): A novel compiler backend that translates DSL constructs into a category-theoretic intermediate representation, enabling rigorous mathematical analysis and transformations.
• Unit-Aware Type System: Robust dimensional analysis and type checking to ensure physical consistency.
• Sheaf-Based Structural Encoding (Planned): Encoding of local laws and symmetries as sections of sheaves over dynamic topological spaces.
• Prototype Engine for Discrete Systems (Planned): Initial simulation and validation of symbolic mass-spring systems.

Future Phases (Conceptual - Under Development)

• Phase 2: Symbolic Physical Representation & PDE Engine: Developing symbolic solvers for field equations and discovering conservation laws.
• Phase 3: Symbolic Theorem Proving and Neural-Symbolic AI: Integrating with formal theorem provers (e.g., Coq, Lean) and training physics-aware transformers for verifiable inference.
• Phase 4: Meta-Learned Control & Game-Theoretic Dynamics: Synthesizing symbolic control policies for multi-agent physical systems with formal guarantees.
• Phase 5: Simulation, Visualization & Deployment: Building an interactive simulation engine, advanced visualization tools, and deploying as an open-source platform.