Co-Optimized Inverse Design Insights

• Co-optimized inverse design is an advanced computational strategy that simultaneously optimizes coupled system variables to meet stringent multi-objective performance metrics.
• It leverages alternating convex optimization and regularized least-squares to ensure physically realizable and manufacturable solutions in complex domains like nanophotonics and self-assembly.
• The integrated framework reduces computational cost and expands design possibilities by jointly optimizing material, geometry, and field responses for high-performance engineered systems.

Co-optimized inverse design is an advanced computational strategy in which two or more coupled aspects of a system—typically physical structure and functional response—are simultaneously optimized to meet stringent, often multi-objective, performance metrics. Unlike conventional inverse design, which often fixes one component and inverts the design mapping for the other, co-optimized approaches use alternating or unified optimization of multiple coupled subproblems (e.g., material distribution and electromagnetic fields, or self and cross interactions in assembly) to achieve globally coherent, physically realizable solutions. This methodology has proven especially effective in nanophotonics, soft robotics, multicomponent assembly, and other fields where complex interdependencies render direct or sequential optimization suboptimal.

1. Principles of Co-Optimized Inverse Design

Co-optimized inverse design is rooted in the recognition that many engineering problems possess strong coupling between distinct functional domains or design variables. In nanophotonics, for example, the electromagnetic field pattern a device supports is not independent of the underlying permittivity landscape, and vice versa. The framework typically casts the design as a set of interleaved or coupled optimization problems, usually of the convex or bi-convex form when the problem structure allows.

A canonical example is provided by the complementary convex optimization strategy for photonic device design (0912.4425). Here, two convex subproblems are solved iteratively:

• Material update: Holding the target field fixed, the dielectric (or inverse dielectric) profile is computed by solving a least-squares or bounded convex optimization.
• Field update: Holding the dielectric profile fixed, the field is adjusted (e.g., via another least-squares problem with additional regularization) to be closer to a realizable field supported by that dielectric.

This interleaving exploits the linearity of Maxwell’s equations with respect to different variables when the others are held fixed. The general pattern recurs in other domains: for multicomponent self-assembly, both self and cross potentials are iteratively tuned based on their individual contributions to the collective target (Piñeros et al., 2018); in neural network–based inverse design, both continuous and discrete variables (e.g., material selection) are optimized jointly in mixed-integer programming (Ansari et al., 2021).

2. Mathematical and Algorithmic Frameworks

Co-optimized methods often adopt one or more of the following algorithmic paradigms:

• Alternating Convex Optimization: Each subproblem (e.g., dielectric update, field update) is convex in isolation and is solved sequentially with updated inputs from the other. In (0912.4425), this yields rapid convergence for 1D and 2D resonator inverse designs, even under physical constraints such as bounds on permittivity.
• Regularized Least-Squares/Constrained Optimization: Physicality or manufacturability is enforced via regularization, hard bounds, or penalty functions. For example, regularization terms like $\eta\|y-y_0\|^2$ control the deviation from an initial guess or enforce binary (fabricable) material distributions.
• Multi-Objective and Multimodal Formulation: Co-optimizing for multiple resonances, polarization states, or field profiles naturally requires multi-objective optimization, often realized by extending least-squares objectives (e.g., as in multi-mode photonic resonator design (0912.4425)) or matching multiple radial distribution functions in multicomponent assembly (Piñeros et al., 2018).
• Hybrid or Hierarchical Optimization: Methods may combine local (gradient-based or adjoint) and global (genetic algorithm, memetic algorithm, or Bayesian) optimization layers, as in memetic inverse design for antennas (Capek et al., 2021) and hybrid ML/topology optimization for nanophotonics (Marzban et al., 23 May 2025).
• Coupled Parameter Updates and Surrogate Model Integration: Neural or differentiable surrogate models may be embedded directly as constraints or regularizers in the optimization loop (Ansari et al., 2021), enabling inclusion of fabrication constraints, combinatorial variables, or multi-domain objectives.

In all cases, the design variable set encompasses all controllable degrees of freedom—structural, material, or interaction potentials—that contribute to the system’s physics.

3. Representative Applications

Co-optimized inverse design frameworks are widely employed in nanophotonics, self-assembled materials, and other domains:

System/Application	Co-Optimized Quantities	Framework/Method
Nanophotonic Resonators	Dielectric profile & field/polarization	Alternating convex optimization (0912.4425)
Multicomponent Self-Assembly	AA, BB (self) & AB (cross) potentials	Relative entropy minimization (Piñeros et al., 2018)
Soft Robotics	Morphology & target dynamical behavior	Energy-based analytical modeling (Osorio et al., 2024)
Antenna and Metamaterial Design	Shape topology & induced currents	Memetic algorithm, MoM (Capek et al., 2021)
Neural/Combinatorial Inverse Design	Continuous design & material selection	MILP/hybrid optimization (Ansari et al., 2021)

In nanophotonic inverse design (0912.4425), the alternating optimization approach allows the automatic discovery of structures that support complex field profiles (e.g., high-Q or multi-mode resonances), with the ability to include strict permittivity bounds yielding binary (fabricable) geometry. In binary particle assembly (Piñeros et al., 2018), simultaneous optimization of self and cross potentials creates new classes of crystals and open frameworks inaccessible via single-component design.

4. Advantages Over Sequential or Uncoupled Approaches

Co-optimized inverse design methods offer several compelling advantages:

• Mitigation of Nonphysical or Nonfabricable Solutions: Strict bounds and regularization (imposed directly in the optimization step) ensure that designs are physically viable and manufacturable, e.g., by promoting binary material grids or excluding singular dielectric features (0912.4425).
• Access to Richer Solution Spaces: Partitioning degrees of freedom (e.g., self/cross interactions in assembly (Piñeros et al., 2018)) enables the mechanism by which combinatorially rich target structures can be realized efficiently, lowering the required potential complexity compared to single-component approaches.
• Reduction of Computational Cost: When the optimization subproblems are convex or otherwise efficiently solvable, convergence is rapid (often in seconds for 1D, minutes for 2D nanophotonic problems (0912.4425)), and the search may avoid the sample inefficiency of brute-force or purely global approaches.
• Flexibility and Modular Extension: The alternating or coupled framework is easily extensible—new performance criteria (e.g., suppression of unwanted Fourier components, specification of mode area/Q, or geometric constraints) can be incorporated by augmenting the regularization or constraint terms in the optimization (0912.4425). Multi-objective (multi-mode or multi-parameter) optimization is handled naturally.
• Improved Convergence and Robustness: By alternating between coupled variables, the co-optimization approach can systematically drive the solution toward a design that jointly satisfies all physical constraints and targeted performance metrics, mitigating local minima that often plague purely sequential or “one-shot” inverse mappings.

5. Scalability, Limitations, and Future Directions

The co-optimized inverse design framework exhibits robust scalability in lower-dimensional problems, and its architecture is, in principle, extensible to higher dimensions and more complex multiphysics problems. However, specific challenges remain:

• Scaling to Large and 3D Systems: The number of degrees of freedom grows rapidly, particularly in 3D nanophotonic or soft robotic designs. While iterative and adjoint or sensitivity-based solvers offer feasible computational paths, memory and time requirements can become prohibitive without problem-specific acceleration.
• Integration of Fabrication and Robustness Constraints: Strict binarization and upper/lower material bounds are effective, but translating soft or process-induced variations (e.g., fabrication tolerances, multistability in soft robotics) into the optimization loop is an active area of research (0912.4425, Capek et al., 2021).
• Generalization Beyond Specific Physics: Although the underlying mathematical decomposition is general (e.g., applicable to acoustics, mechanics), each domain may require bespoke discretization and regularization strategies.
• Multi-Objective and Multi-Physics Co-Design: Simultaneous co-optimization of several physical attributes (e.g., photonic and mechanical properties, or field/interaction potential and geometry) is enabled by extending the optimization framework to include multiple coupled variables and constraints.
• Synergy with Machine Learning and Surrogate Models: Modern extensions embed differentiable and physics-informed neural surrogates directly in the optimization loop (Ansari et al., 2021), enabling gradient-based co-optimization even in highly nonlinear or discontinuous design spaces.

6. Impact and Broader Significance

Co-optimized inverse design marks a paradigm shift for the computational discovery and realization of high-performance engineered systems. Its core premise—that multiple, tightly coupled aspects (material, geometry, field, control) must be optimized in an integrated fashion—has enabled breakthroughs in:

• High-Q and multi-modal nanophotonic resonators, supporting new regimes of light-matter interaction and miniaturized optical devices.
• Multicomponent material self-assembly, yielding a broader spectrum of possible microstructures with designable mechanical, optical, or electromagnetic properties.
• Fabrication-aware device engineering, with solutions tailored from the outset to the constraints of process or assembly.
• Multistable and “embodied-intelligence” soft robots, where morphology and function are coupled to simplify control and expand function (Osorio et al., 2024).

Future directions are likely to focus on further modularity, the integration of uncertainty quantification, real-time or adaptive optimization strategies, and expanding the reach of co-optimization toward full multiphysics design, robust experimental automation, and broader deployment in materials and device discovery pipelines.