Inverse-Designed Surrogate Networks

• Inverse-designed surrogate networks are efficient models that replace costly forward simulations with neural or hybrid approximations for inverse design and uncertainty quantification.
• They integrate physics-informed techniques like FE-PINNs, theory-guided auto-encoders, and differentiable simulators to enforce physical consistency while accelerating optimization and parameter inference.
• Invertible and probabilistic architectures resolve non-uniqueness in inverse problems by modeling multimodal solution distributions, enhancing accuracy for applications in photonics, materials, and multiphase flows.

Inverse-designed surrogate networks are surrogate models used for inverse design, inverse modeling, and Bayesian inverse problems, where a neural or hybrid surrogate replaces an expensive forward code, approximates the relationship between model parameters and responses, directly learns an inverse surrogate model, or provides the full distribution of possible solutions to the inverse design problem (Wang et al., 2020, Padmanabha et al., 2020, Frising et al., 2022). Recent formulations include finite-element-based physics-informed neural networks, theory-guided auto-encoders, differentiable graph neural network simulators, invertible neural networks, probabilistic surrogate networks, and surrogate scattering-matrix workflows for photonic processors (Sunil et al., 2024, Choi et al., 2024, Munk et al., 2019, Muda et al., 23 Apr 2026). The literature indicates that these systems need not be direct inverse maps: they also include forward surrogates embedded in optimization loops, joint forward-and-inverse models, and probabilistic samplers over non-unique solution sets.

1. Problem setting and mathematical objectives

In Bayesian inverse problems, a common setting is the direct model $y=f(x)+\varepsilon$, with posterior inference targeting $p(x\mid \mathbf{y})$ under expensive forward evaluation (Lartaud et al., 2024). The practical difficulty is that only a small region of the input space is relevant, so space-filling designs or classic D/I-optimal designs can waste computational budget outside the region of interest (Lartaud et al., 2024). In time-dependent inverse UQ, the quantity-of-interest may be an infinite-dimensional response, which motivates dimensionality reduction before surrogate construction (Xie et al., 2023).

A central difficulty is non-uniqueness. Inverse problems often need to learn a one-to-many non-linear mapping, and comparable or even identical performance may be realized by different designs, yielding a multimodal distribution of possible solutions to the inverse design problem (Yang et al., 2021, Frising et al., 2022). This is explicit in inverse photonic design, where symmetry-induced multimodality causes standard networks trained with MSE to average distinct solutions and produce outputs that correspond to no true device (Frising et al., 2022). It also appears in scientific inversion from sparse and noisy observations, where high-dimensional spatial fields are inferred from indirect measurements under ill-posedness and incomplete data (Padmanabha et al., 2020).

A further criterion is deployment fidelity. Autoinverse states that the most important property of a successful neural inverse method is the performance of its solutions on the native forward process, and not only on the learned surrogate (Ansari et al., 2022). This emphasis separates inverse-designed surrogate networks from purely self-consistent inverse models: accuracy on the surrogate is insufficient if the solution lies in an unreliable or out-of-distribution region.

2. Physics-guided and differentiable forward surrogates

A major class of inverse-designed surrogate networks learns the forward physics while preserving the governing structure. FE-PINNs use the finite element method to train physics-informed neural networks suitable for surrogate modeling, with loss based on the residual of the FEM weak formulation rather than pointwise PDE residuals (Sunil et al., 2024). The core loss is

$\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$

where $R(\tilde{d})$ is the finite-element residual obtained by inserting the network prediction into the FE equations $G(\tilde{d})=0$, and for linear problems $G(d)=Kd-F_e$ (Sunil et al., 2024). Because the FEM weak form encodes Dirichlet and Neumann boundary conditions directly in the governing equations, boundary conditions are automatically enforced in the weak sense during training (Sunil et al., 2024). The custom stencil convolution uses the inverse isoparametric map and FE shape functions to execute convolution-like operations directly on unstructured FEM meshes, with locality, weight sharing, and multi-scale representation (Sunil et al., 2024). In the reported tests, testing errors systematically decrease as the number of training geometries is increased, and the testing error decreases monotonically as more diverse geometries are included in training (Sunil et al., 2024).

Theory-guided Auto-Encoder models embed discretized governing equations into training. TgAE is built on a convolutional auto-encoder architecture, and the residual of the discretized governing equations as well as the data mismatch constitute the loss function (Wang et al., 2020). Its total loss is

$$\mathcal{L}(\theta)=\mathcal{L}_{\text{data}}(\theta)+\mathcal{L}_{\text{PDE}}(\theta)+\mathcal{L}_{\text{BC}}(\theta)+\mathcal{L}_{\text{IC}}(\theta),$$

with the PDE residual computed from a finite-difference discretization of the governing equation (Wang et al., 2020). The framework is used for surrogate construction, uncertainty quantification, and inverse modeling, and the reported results show satisfactory accuracy with high $R^2$, low $L_2$ errors, speedups of 1–2 orders of magnitude in uncertainty quantification tasks, and satisfactory results in parameter inversion (Wang et al., 2020).

Differentiable simulators extend this logic from residual-based training to rollout-based optimization. The differentiable graph neural network simulator for granular flows represents the system as a graph, predicts the evolution of the graph at the next time step, and uses reverse mode automatic differentiation with backpropagation through time for inverse estimation (Choi et al., 2024). Because the simulator is differentiable, gradient-based optimization becomes applicable for material parameter estimation, boundary-condition inference, and baffle design, and the reported framework offers an orders of magnitude faster solution than the conventional finite difference approach to gradient-based optimization, including a 151x speedup over MPM+FD in a single-parameter case (Choi et al., 2024). In a related high-energy-density setting, a causal, dynamic, multifidelity reduced-order surrogate maps radiation temperature drive to DT interface radius and velocity dynamics by learning a controller for a base analytical model, integrating operator learning, causal architectures, and physical inductive bias (Maltba et al., 5 Sep 2025).

3. Invertible, generative, and structure-preserving inverse surrogates

Normalizing-flow and invertible architectures address non-uniqueness by modeling conditional solution distributions rather than a single inverse map. Inverse photonic design with conditional invertible neural networks uses the conditional mapping $z=f(x;y)$ and $p(x\mid \mathbf{y})$0, trained by maximum likelihood, to provide the full distribution of possible solutions to the inverse design problem and resolve ambiguity in nanodevice inverse design problems featuring multimodal distributions (Frising et al., 2022). The negative log-likelihood objective is

$p(x\mid \mathbf{y})$1

and the paper reports superior flexibility and accuracy relative to a cVAE baseline when dealing with multimodal device distributions (Frising et al., 2022).

A broader inverse-surrogate formulation appears in multiphase flow, where a conditional invertible neural network directly learns an inverse surrogate model from sparse and noisy observations to a high-dimensional non-Gaussian permeability field (Padmanabha et al., 2020). The conditional density is written as

$p(x\mid \mathbf{y})$2

and the reported two- and three-dimensional models generate diverse sample realizations whose predictive mean is close to the ground truth even when trained with limited data (Padmanabha et al., 2020). In laser-wakefield acceleration, an invertible neural network surrogate jointly approximates the forward simulation and reconstructs experimentally acquired diagnostics; the reported forward metrics are MSE $p(x\mid \mathbf{y})$3 and SSIM $p(x\mid \mathbf{y})$4, and validation relative errors are $p(x\mid \mathbf{y})$5 for $p(x\mid \mathbf{y})$6, $p(x\mid \mathbf{y})$7 for $p(x\mid \mathbf{y})$8, and $p(x\mid \mathbf{y})$9 for $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$0 (Bethke et al., 2021).

Generative inverse models also operate through latent-space factorization. The GAN+MDN framework for microstructural materials design learns a generator $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$1 from latent vectors to high-dimensional images and an MDN for the conditional density

$\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$2

so that a target property first yields multiple plausible latent codes and then multiple plausible microstructures (Yang et al., 2021). The reported method overcomes one-to-many mapping, low-to-high dimensional inversion, and computational cost, and produces multiple promising solutions in an efficient manner (Yang et al., 2021).

Structure-preserving probabilistic surrogates generalize this idea to simulators with stochastic control flow. Probabilistic Surrogate Networks retain the interpretable structure and control flow of the reference simulator, target cases where the number of random variables is itself stochastic and potentially unbounded, and dynamically expand when new address transitions are encountered during training (Munk et al., 2019). Reported examples include exact replication of posterior distributions and address transition probabilities for an unbounded-randomness program, 99.6% valid programs generated over 50,000 samples in a program-synthesis stack example, and order-of-magnitude speedups in amortized inference for composite-material curing, with factors of 15–90x in traces per second (Munk et al., 2019). For structured outputs, learned differentiable surrogate losses define

$\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$3

learn the embedding $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$4 by contrastive learning, and decode by projected gradient descent so that the method can predict new structures of the output data via a decoding strategy based on gradient descent (Yang et al., 2024).

4. Representations over geometry, meshes, graphs, and scattering operators

Representation choice is a defining axis of inverse-designed surrogate networks. Discretization-independent surrogate modeling over complex geometries uses a main network that consumes pointwise spatial information and returns a continuous representation, so predictions can be queried at any location in the domain (Duvall et al., 2021). Geometry is encoded implicitly through a minimum-distance function,

$\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$5

which is concatenated with spatial coordinates, while geometric design variables are either additional main-network inputs in DV-MLP or hypernetwork inputs in DV-Hnet and NIDS (Duvall et al., 2021). In the reported vehicle-aerodynamics problem with limited training data, the design-variable hypernetwork performs best and the model predictions are obtained many orders of magnitude faster than the full order models (Duvall et al., 2021).

Mesh-aware surrogates use different mechanisms. FE-PINNs execute stencil convolution directly on arbitrary FEM meshes, and the paper hypothesizes that stencil convolution is less sensitive than graph neural networks to mesh connectivity (Sunil et al., 2024). The differentiable granular-flow GNS uses a graph $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$6, with encoder, processor, decoder, and explicit Euler update to model full runout evolution rather than only a low-dimensional input-output map (Choi et al., 2024). This suggests that geometry handling in inverse-designed surrogate networks spans at least three distinct regimes: implicit continuous coordinates, mesh-local stencil operators, and interaction graphs.

In nanophotonics, the surrogate itself may be an operator. Surrogate scattering-matrix inverse design introduces a two-stage workflow in which the trainable optical block is first represented as a passive complex matrix $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$7 with bounded singular values, and the task is solved directly in matrix space at negligible cost (Muda et al., 23 Apr 2026). The selected target operator is then transferred to a fabrication-aware freeform device through an adjoint problem driven by a Frobenius-norm transmission residual and a reflection penalty (Muda et al., 23 Apr 2026). The framework also introduces a banded-router architecture composed with a fixed evanescent-coupling region, exploiting the bandwidth-additive property of matrix products to realize dense effective operators within a design region roughly half as long as a fully local router would require (Muda et al., 23 Apr 2026). On MedMNIST, the realized all-optical classifier reproduces the surrogate accuracy within $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$8 percentage points after only 20 adjoint epochs; on RSSCN7, the banded router plus evanescent stage improves test accuracy by more than 15 percentage points over a linear readout baseline; and a Yin-Yang task confirms support for nonlinear decision boundaries (Muda et al., 23 Apr 2026).

5. Acquisition, adaptation, and optimization around the surrogate

Inverse-designed surrogate networks are often valuable not only as predictors but as objects around which acquisition and optimization are organized. In Bayesian inverse problems, sequential design under the Stepwise Uncertainty Reduction framework selects each new design point by minimizing the expectation of an uncertainty functional after the yet unknown new data point is observed (Lartaud et al., 2024). CSQ adapts D-optimal design by restricting the search to a Mahalanobis ball around the MAP estimate, is computationally efficient in higher dimensions, and empirically outperforms D- and I-optimal designs in banana-shaped and bimodal posterior examples, but does not have supermartingale convergence guarantees (Lartaud et al., 2024). IP-SUR instead uses a posterior-weighted IMSPE criterion, has no user hyperparameters, requires MCMC at each stage, and comes with a theoretical guarantee for the almost sure convergence of the uncertainty functional (Lartaud et al., 2024).

Adaptive multi-fidelity refinement addresses the mismatch between prior-trained surrogates and posterior concentration. The adaptive DNN framework for large-scale Bayesian inverse problems first constructs an offline prior-based DNN surrogate, then locally refines it online by training a shallow network that takes $\mathcal{L}(\tilde{d})=\Vert R(\tilde{d})\Vert,$9 as input, producing a composite multi-fidelity neural network (Yan et al., 2019). In the reported 9-dimensional example, 110 offline and 180 online high-fidelity runs achieve posterior accuracy similar to conventional MCMC with 50,000 forward model evaluations, and in a 111-dimensional problem the adaptive method obtains accurate posterior means and standard deviations with only 950 online forward evaluations versus 50,000 for MCMC (Yan et al., 2019).

Optimization with surrogates can be greedy, gradient-based, or candidate-branching. ALPS uses a Random Forest forward surrogate, PCA-compressed spectra, Latin Hypercube Sampling, and a greedy prediction-based exploration strategy that repeatedly selects the batch with the smallest predicted discrepancy to a target optical curve (Grbcic et al., 2024). It exhibits superior performance when compared to other optimization algorithms across all benchmarks, and warm starting for changed target optical characteristics further enhances performance (Grbcic et al., 2024). In artificial electromagnetic materials design, the neural adjoint method and the Neural Lagrangian extension make explicit that a more accurate surrogate simulator leads to better solutions (Fujii et al., 2023). NeuLag prunes to promising candidates, branches them through a learned projector,

$R(\tilde{d})$0

and then continues optimization; the reported resimulation errors are approximately $R(\tilde{d})$1 compared to previous methods for three AEM tasks, while retaining support for soft and hard constraints (Fujii et al., 2023).

6. Uncertainty, feasibility, consistency, and limitations

Uncertainty is both a modeling target and a regularizer. Autoinverse estimates aleatoric and epistemic uncertainty with deep ensembles and modifies inversion by minimizing target mismatch together with both uncertainty terms,

$R(\tilde{d})$2

thereby seeking inverse solutions in the vicinity of reliable data sampled from the forward process (Ansari et al., 2022). The method is reported to enforce feasibility, come with embedded regularization, and be initialization free (Ansari et al., 2022). In transient inverse UQ, functional PCA and Bayesian neural networks separate phase and amplitude variability before surrogate learning, and only methods with code uncertainty produce credible intervals that consistently envelop experimental data; the full fPCA + BNN pipeline shows better agreement with the experimental data in both in-sample and out-of-sample settings (Xie et al., 2023).

Consistency constraints provide another route to robust inversion. The MaCC framework trains surrogates that are manifold-consistent, so predictions are always physically meaningful, and cyclically consistent, so a forward prediction followed by an independently trained inverse model returns the original input parameters (Anirudh et al., 2019). The reported surrogates have lower MSE on image outputs, are more resilient to sampling artifacts, and tend to be more data efficient, with improvements of up to 30% in small-data regimes (Anirudh et al., 2019). This suggests that inverse-designed surrogate networks often benefit from explicit constraints on either uncertainty, physical validity, or forward-inverse agreement rather than relying on data fit alone.

Several limitations recur across the literature. FE-PINNs trained on one geometry generalize reasonably only to similar geometries, and for distinct geometries errors grow; training on more diverse geometries improves interpolation across the range represented in the training set (Sunil et al., 2024). The cINN inverse surrogate for multiphase flow is limited to the broad prior distribution with which it is trained, and out-of-prior generalization is not guaranteed (Padmanabha et al., 2020). IP-SUR has convergence guarantees but is computationally heavier because it requires MCMC at each stage, whereas CSQ is faster but lacks formal proof (Lartaud et al., 2024). Large and accurate surrogate simulators can reduce feasible batch size and make optimization over sufficient candidates difficult, a point emphasized in the transition from NA to NeuLag (Fujii et al., 2023). In photonic neural networks, direct geometry-to-task pipelines scale the number of electromagnetic simulations with both network depth and batch size, whereas the two-stage matrix-to-device workflow removes the minibatch dependence from the full-wave loop and reduces simulation budgets by orders of magnitude (Muda et al., 23 Apr 2026).