Physics-Constrained Optimization
Physics-constrained optimization searches for the best design while requiring that governing equations and physical limits are respected.
Optimizing subject to physics
Engineering design is optimization: adjust parameters to minimize or maximize an objective. What distinguishes physics-constrained optimization is that the search must obey the laws of physics and hard operating limits at every candidate. A design that violates conservation, exceeds a stress limit, or is physically unrealizable is not a valid point in the search, no matter how good its objective.
Enforcing the constraints
- Equality constraints: the state must satisfy the governing equations
- Inequality constraints: temperatures, stresses, or fields must stay within limits
- Penalty methods: add a cost for violations, giving soft enforcement
- Projection or barrier methods: keep every iterate inside the feasible region
Gradients through the physics
Efficient optimization needs the gradient of the objective and constraints with respect to the design parameters. When the constraints are simulations, these gradients come from adjoint methods or differentiable simulators, which supply the full sensitivity at a cost independent of the number of parameters. This is what makes optimization over high-dimensional designs tractable.
The role of surrogates
When each physics evaluation is expensive, a surrogate can stand in during the search, with the true solver used to check promising candidates. Surrogate-based optimization, guided by uncertainty-aware acquisition rules, finds good designs with few expensive runs. The surrogate's uncertainty also warns when the search strays into regions where it cannot be trusted.
Application to fusion design
Designing a confinement device is a physics-constrained optimization: shape, field, and profile parameters are tuned to meet performance targets while satisfying equilibrium, stability, and engineering limits. For the Hyperion breeder, a spherical tokamak, design studies explore such trade-offs in simulation, with parameters like plasma current near 9.86 MA emerging from constrained searches rather than being chosen freely. These are computational design results, not measurements from built hardware.