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AI for Simulation

This section collects methods and design patterns to integrate learning-based components into a physics simulation stack.

The focus is on mechanics / PDE-inspired discretizations (meshes, graphs, finite elements) and on workflows compatible with implicit solvers and differentiation through simulation.

Why AI in a constrained differentiable solver?

In an implicit constrained setting, AI can be useful without replacing physics:

  • Reduce Newton iterations: Better initial guesses or learned correction directions can significantly reduce the number of nonlinear iterations required for convergence
  • Accelerate Krylov solvers: Learned preconditioners, low-rank updates, and learned coarse spaces can dramatically improve linear solver convergence
  • Learn difficult-to-model components: Constitutive laws surrogates can be learned from data while preserving the overall solver structure
  • Parameter optimisation: Apply automatic differentiation directly to parameters to be optimized, leveraging the end-to-end differentiability of the solver

This approach preserves:

  • Exact constraint enforcement via Lagrange multipliers
  • Energy-conserving properties of implicit schemes
  • Convergence guarantees
  • End-to-end differentiability for inverse problems

Connection to SOFAx structure

The SOFAx framework is designed to enable AI integration at multiple levels:

  1. Residual evaluation: The residual operator can incorporate learned constitutive laws as pure functions, with automatic differentiation providing the required tangent operators

  2. Linear operator actions: Matrix-free JVP/VJP operations enable learned preconditioners to be inserted as operators without matrix assembly

  3. Iterative solver structure: The Newton–Krylov solver provides natural hooks for warm starts, correction predictions, and approximate inverse actions

  4. Geometric decomposition: The constraint structure described in Constraints & Physical Interactions enables learned components to respect the \(\ker G \oplus \operatorname{range}(G^\top)\) decomposition

  5. Schur complement: The physical interpretation of the Schur complement (see Preconditioning) as dynamic compliance motivates learned approximations that respect the underlying physics