Multiphysics Inverse-Design Algorithm

• The multiphysics inverse-design algorithm is defined by coupling design parameters, forward physics models, and iterative optimization to achieve target functionalities.
• It spans formulations from discrete topology and level-set methods to PDE-constrained and latent-space optimization, balancing diverse physical effects.
• The method is applied in antenna, optomechanical, and metamaterials design, emphasizing constraint handling, robustness, and data efficiency.

Searching arXiv for papers on multiphysics inverse design algorithms to support the article. A multiphysics inverse-design algorithm is a computational framework that determines a geometry, topology, material distribution, or parameter set from prescribed performance targets by coupling a design representation to one or more forward physics models and an optimization loop. In recent arXiv literature, the term spans discrete topology optimization with binary material layouts, level-set and adjoint-state formulations for PDE-constrained dynamics, reduced-order circuit-network models for electrically large metastructures, multi-fidelity workflows that ration high-fidelity simulations, and latent-space generative/operator models that decode both structure and governing-field solutions from a shared low-dimensional variable (Kadlec et al., 2024, Voronov et al., 2024, Szymanski et al., 2021, Grbcic et al., 2023, Zang et al., 10 Sep 2025).

1. Conceptual scope

Across the literature, a multiphysics inverse-design algorithm is defined less by a single optimizer than by a recurring architecture: a design parameterization, a forward map from design to one or more physical fields, an objective or constraint set derived from those fields, and an update rule that modifies the design until the target functionality is achieved. In some formulations the physics are explicitly coupled, as in resonant optomechanics where optical and mechanical eigenmodes jointly determine the vacuum optomechanical coupling $g_0$ and the scattering rate $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$; in others, one geometry simultaneously controls several effective properties, as in thermo-mechanical inverse homogenization of cellular materials (Hambraeus et al., 5 May 2026, Gavazzoni et al., 2022).

The literature also uses “multiphysics” across a spectrum. One end is fully coupled PDE systems, such as micromagnetic dynamics with exchange, demagnetizing field, external excitation, and geometry-dependent material distribution, or optomechanical cavities with coupled Maxwell and elastodynamic eigenproblems (Voronov et al., 2024, Hambraeus et al., 5 May 2026). The other end is a shared-geometry, multi-model workflow in which different solvers, fidelities, or reduced models are coordinated inside one inverse-design loop, as in plasma metamaterials with Drude dispersion, quartz envelopes, non-uniform density profiles, and robustness analysis, or in multi-fidelity optimization that combines low-fidelity machine learning screening with high-fidelity simulation calls (Rodriguez et al., 2022, Grbcic et al., 2023). This suggests that the defining feature is not a specific numerical method, but the joint treatment of multiple physical effects, constraints, or solver layers within a single optimization pipeline.

2. Mathematical formulation

A common starting point is a PDE-constrained or topology-constrained optimization problem. In discrete topology optimization, a design domain $\Omega_0$ is discretized into $N_{\mathrm{el}}$ cells and encoded by a binary topology vector $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$, leading to a multi-objective problem of the form

$$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$

with local scalarization

$$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$

The objective components can mix physical metrics and topological regularizers, such as $Q$, realized gain, reflection coefficient, electrical size, area, smoothness, or manufacturability metrics (Kadlec et al., 2024).

A second, more explicitly PDE-constrained formulation writes the physics as

$\mathcal{N}(u,\mu)=s,\qquad \mathcal{B}(u)=g,$

where $u$ is a field or set of fields, $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$0 is a design-dependent coefficient field, and $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$1 and $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$2 encode source and boundary data. Design-GenNO uses exactly this form for two-phase materials, with $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$3 as the microstructure and $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$4 as the PDE solution, then performs inverse design in a learned latent space $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$5 that decodes both $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$6 and $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$7 (Zang et al., 10 Sep 2025). Level-set micromagnetic optimization adopts the same pattern with a design vector of radial-basis amplitudes $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$8, an implicitly defined interface $\Gamma_{\mathrm{OM}}=4g_0^2/\kappa$9, and forward dynamics $\Omega_0$0 governed by the Landau–Lifshitz–Gilbert equation (Voronov et al., 2024).

Another recurrent formulation uses a single objective with multiple physics-based inequality constraints. In thermo-mechanical inverse homogenization, the density field $\Omega_0$1 minimizes the unit-cell mass

$\Omega_0$2

subject to periodic elasticity and heat-conduction corrector problems and bounds on homogenized stiffness and conductivity components and ratios, including $\Omega_0$3, $\Omega_0$4, $\Omega_0$5, $\Omega_0$6, and $\Omega_0$7 (Gavazzoni et al., 2022). The choice between vector objectives, scalarized objectives, and constrained formulations is therefore a central modeling decision, not an implementation detail.

3. Design representations and forward models

The design variable is the point where inverse-design algorithms differ most strongly. Recent work uses at least five distinct representations: binary topology vectors, level-set functions, parameterized shapes, density fields, and learned latent variables. The associated forward models range from direct PDE solvers to reduced-order operator surrogates.

Representation	Forward model	Typical use
$\Omega_0$8	MoM, stiffness/thermal matrices, rank-1 updates	Discrete topology search (Kadlec et al., 2024)
Level-set $\Omega_0$9 from RBF amplitudes $N_{\mathrm{el}}$0	LLG in NeuralMag with adjoint ODE	Micromagnetic topology optimization (Voronov et al., 2024)
Parameterized geometry $N_{\mathrm{el}}$1	Maxwell and elastodynamic eigenproblems in COMSOL	Resonant optomechanics (Hambraeus et al., 5 May 2026)
Cellwise density $N_{\mathrm{el}}$2	Periodic elasticity and heat-conduction homogenization	Thermo-mechanical cellular materials (Gavazzoni et al., 2022)
Latent variable $N_{\mathrm{el}}$3 or encoded field $N_{\mathrm{el}}$4	MultiONet, NF priors, AE/VAE decoders, ID-DMD	Microstructure and field inversion (Zang et al., 10 Sep 2025, Lim et al., 25 Feb 2026, Zhu et al., 13 Feb 2025)

Binary topology is used when topology changes are themselves the design act. In antenna optimization, flipping one bit corresponds to adding or removing one cell, and the resulting system matrix change is treated as rank-1 or low-rank, which supports extremely fast local updates (Kadlec et al., 2024). In inverse-design magnonics, the physical mask is likewise binary—YIG or void—but the optimizer uses a modified direct binary search rather than a memetic multi-objective scheme (Wang et al., 2020). In planar THz filters, the chromosome is a 2D binary Au/air pattern whose columns are mapped to coplanar-stripline sections with local impedance $N_{\mathrm{el}}$5 and phase constant $N_{\mathrm{el}}$6, and the forward model is an ABCD cascade instead of a full-wave PDE solve (Dehghanian et al., 3 Jun 2025).

Level-set and parametric-shape methods replace explicit binary flipping with differentiable geometry maps. In the micromagnetic level-set framework, cone-like RBFs define $N_{\mathrm{el}}$7, a sigmoid $N_{\mathrm{el}}$8 converts $N_{\mathrm{el}}$9 into smooth material coefficients, and NeuralMag supplies differentiable exchange and demagnetizing operators on GPU (Voronov et al., 2024). In release-free optomechanical crystals, the design remains a structured line-defect crystal with elliptical holes, but roughly $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$0 geometric variables in the defect and transition region are optimized because the figure of merit depends jointly on optical frequency and $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$1, mechanical frequency and $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$2, and optomechanical overlap terms $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$3 and $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$4 (Hambraeus et al., 5 May 2026).

A third class replaces the direct field solve with a learned or reduced operator. Plasma metamaterial inverse design still solves TM-polarized Maxwell equations in frequency domain, but the constitutive law already couples electromagnetics to a Drude plasma model with collisionality, a sixth-order radial density profile, and quartz envelopes (Rodriguez et al., 2022). Design-GenNO goes further by decoding full PDE solution fields from a latent variable through MultiONet and regularizing the latent distribution with a normalizing flow (Zang et al., 10 Sep 2025). ID-DMD constructs a low-rank parametric state-space model,

$\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$5

or its Koopman-lifted variant, and then performs inverse design on the reduced operator rather than on the original PDE solver (Zhu et al., 13 Feb 2025).

4. Optimization architectures

Recent multiphysics inverse-design algorithms fall into four broad optimization families: adjoint-state gradient descent, hybrid global–local memetic search, population-based evolutionary search, and latent-space or reduced-model optimization. The adjoint family is dominant when the forward model is differentiable and the number of design variables is large. In micromagnetics, the adjoint variable $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$6 is evolved backward in time, and the design gradient is recovered from

$\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$7

which permits memory-efficient optimization through torchdiffeq and avoids storing the entire magnetization history (Voronov et al., 2024). In resonant optomechanics, discrete adjoints for eigenvalues and continuous adjoints for eigenvectors are combined so that one can optimize a figure of merit built from $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$8, $\mathbf{g}\in\{0,1\}^{N_{\mathrm{el}}}$9, $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$0, frequencies, and fabrication penalties using ADAM (Hambraeus et al., 5 May 2026). In metastructured devices, the forward problem is a sparse linear network system and the adjoint variable method reduces gradient cost from $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$1 forward solves to $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$2 linear solves for $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$3 scenarios, independent of the number of design variables (Szymanski et al., 2021).

Hybrid global–local schemes are used when the design space is discrete or strongly multimodal. MOMA-AW combines NSGA-II in objective space with a rank-1 perturbation local optimizer in topology space; each agent carries its own adaptive weight vector on the simplex, and local search minimizes a scalarized $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$4 while NSGA-II preserves non-dominated diversity (Kadlec et al., 2024). The same broad pattern appears in GLOnet, except that the search distribution is represented by a generator $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$5, and forward and adjoint electromagnetic simulations provide the efficiency and efficiency gradient used to train the generator distribution directly (Jiang et al., 2019). The genetic-algorithm THz filter framework also belongs here, but with a reduced electromagnetic objective and no adjoint: a rank-based tournament selection, two-point crossover, and bit-flip mutation operate on a binary CPS mask, while fitness is the RMSE between target and simulated $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$6-parameter magnitude and phase (Dehghanian et al., 3 Jun 2025).

The most reduced architectures push the optimization itself into a learned subspace. ID-DMD performs inverse design on a low-rank parametric operator and reports an order of magnitude more accuracy than competing methods while being $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$7–$$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$8 orders faster on several engineering design problems (Zhu et al., 13 Feb 2025). Design-GenNO optimizes $$\min_{\Omega_n \subseteq \Omega_0} \mathbf{f}(\Omega_n) = \big[f_1(\Omega_n),\dots,f_M(\Omega_n)\big]^{\mathrm T},$$9 directly using differentiable objectives built from decoded fields and a latent prior, which turns the original high-dimensional inverse design into a smooth latent optimization problem (Zang et al., 10 Sep 2025). This suggests a useful taxonomy: some algorithms search in physical space and compress the solver, while others compress the design space itself.

5. Constraints, robustness, and data efficiency

Constraint treatment is a central part of multiphysics inverse design because the best unconstrained designs are often physically unusable. The recent literature uses both hard and soft mechanisms. Soft constraints appear as objective terms or penalties: regularity metrics such as

$$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$0

in antenna topology optimization, size and centering penalties in level-set micromagnetics, and weighted frequency or $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$1-penalties in optomechanical cavity design (Kadlec et al., 2024, Voronov et al., 2024, Hambraeus et al., 5 May 2026). Hard constraints appear through design parameterization and morphology: connectivity-preserving mutation in planar THz filters, fixed rod/quartz geometry and bounded plasma frequency in plasma metamaterials, and erosion–dilation operators that remove sub-$$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$2 nm bridges and enforce curvature radius $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$3 nm in release-free optomechanical crystals (Dehghanian et al., 3 Jun 2025, Rodriguez et al., 2022, Hambraeus et al., 5 May 2026).

Robustness is treated in several distinct ways. Plasma metamaterial devices are post-processed with stochastic density perturbations rather than with an explicit robust objective, showing functionality up to perturbation factors $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$4 for waveguides and $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$5 for demultiplexers (Rodriguez et al., 2022). Optomechanical crystals use random $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$6 nm erosion/dilation per iteration to improve robustness against uniform over- or under-etch and then evaluate sidewall-angle sensitivity separately (Hambraeus et al., 5 May 2026). Inverse-designed magnonic devices are checked against 10 nm over-etch, smaller simulation cells, and thermal noise at $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$7 K, while the functionality remains recognizable (Wang et al., 2020). The common pattern is that robustness is rarely an afterthought; it is embedded either in the search operators, the geometry map, or the objective.

Data efficiency is increasingly treated as part of the algorithm rather than merely as a dataset issue. MOMA-AW is explicitly described as an efficient data miner for machine learning because one run produces hundreds of non-dominated, locally optimal designs with associated performance tuples (Kadlec et al., 2024). Multi-fidelity optimization trains an ML model on low-fidelity data to both compress the search-space boundaries before optimization and decide, within each optimization cycle, whether a high-fidelity simulation is warranted (Grbcic et al., 2023). Design-GenNO introduces virtual observations based on weak residuals so that unlabeled microstructures and even self-supervised training can contribute to learning the structure–property operator (Zang et al., 10 Sep 2025). ID-DMD approaches the same problem from the opposite direction: it compresses multiple experiments into a low-rank parametric dynamical model so that inverse design can run on laptop-level computing (Zhu et al., 13 Feb 2025).

6. Applications, misconceptions, and directions

The application range is already broad. Antenna design uses adaptive multi-objective memetic optimization over binary layouts (Kadlec et al., 2024). Magnonics employs both direct binary search on hole patterns and level-set adjoint optimization for devices such as demultiplexers, nonlinear switches, circulators, and frequency-selective spin-wave routers (Wang et al., 2020, Voronov et al., 2024). Plasma metamaterials optimize realistic discharge-tube arrays under Drude loss, quartz envelopes, and perturbation sensitivity (Rodriguez et al., 2022). Release-free optomechanical crystals co-optimize optical and mechanical eigenmodes to increase $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$8 while preserving thermal anchoring (Hambraeus et al., 5 May 2026). Thermo-mechanical inverse homogenization designs lightweight cellular materials with prescribed $$\widetilde f(\Omega_n)=\mathbf{w}^{\mathrm T}\mathbf{f}(\Omega_n)=\sum_{m=1}^M w_m f_m(\Omega_n), \qquad \sum_m w_m=1,\; w_m\ge 0.$$9 and $Q$0 tensors (Gavazzoni et al., 2022). Guided-wave THz filters, MIMO metastructured devices, and inverse microstructure generation extend the same logic to transmission-line, circuit-network, and materials contexts (Dehghanian et al., 3 Jun 2025, Szymanski et al., 2021, Zang et al., 10 Sep 2025).

A persistent misconception is that multiphysics inverse design is synonymous with adjoint topology optimization on a continuous density field. The literature does not support that restriction. Discrete bit-flip search, level-set parameterization, parametric-shape optimization, reduced-order circuit models, binary genetic algorithms, latent generative models, and operator-learning frameworks all qualify when they couple design variables to multiple physical effects or to shared multiphysics objectives (Kadlec et al., 2024, Szymanski et al., 2021, Dehghanian et al., 3 Jun 2025, Zang et al., 10 Sep 2025). Another misconception is that every useful inverse-design loop must place the highest-fidelity solver at every iteration. Reduced-order network solvers, ABCD cascades, low-rank DMD models, and multi-fidelity screening show the opposite: much of the recent progress lies precisely in rationing expensive simulations without losing the relevant design physics (Szymanski et al., 2021, Dehghanian et al., 3 Jun 2025, Grbcic et al., 2023, Zhu et al., 13 Feb 2025).

The main limitations are equally clear. Global optimality is generally not guaranteed for gradient-based or local discrete search; several papers explicitly rely on multiple initializations, adaptive weight distributions, or human-guided starting points to mitigate local minima (Voronov et al., 2024, Kadlec et al., 2024, Hambraeus et al., 5 May 2026). Approximation regimes remain important: 2D TM-only plasma models, linearized material laws, small-angle or weakly nonlinear regimes, and radiation-limited rather than fabrication-limited $Q$1-optimization all delimit what is optimized (Rodriguez et al., 2022, Hambraeus et al., 5 May 2026). Current directions therefore emphasize joint latent spaces for coupled physics, PINNs or neural operators for coupled PDEs, geometry-independent representations, and multiphysics co-designs such as opto-mechanical, thermo-optic, electro-optic, piezo-optomechanical, and electro-optomechanical systems (Marzban et al., 1 Jul 2025, Hambraeus et al., 5 May 2026). This suggests that the field is moving toward reusable inverse-design platforms in which representation learning, differentiable physics, and robust constraint handling are combined rather than treated as separate algorithmic choices.