POD–DeepONet Surrogate Modeling

• The paper presents a hybrid framework that couples proper orthogonal decomposition with DeepONet, reducing high-dimensional operator learning errors.
• It employs a fixed or trainable POD trunk with neural branches that predict modal weights, enabling flexible forward, inverse, and multi-fidelity modeling.
• Quantitative results demonstrate substantial error reduction and speedup, validating the approach for applications in fluid dynamics, photonics, and poroelasticity.

A POD–DeepONet surrogate is a reduced-order operator learning framework that couples proper orthogonal decomposition (POD) low-rank projections with deep operator networks (DeepONets), enabling accurate, efficient emulation of complex physics while compressing the parameter-space and providing interpretable modal decompositions. In this architecture, the POD basis extracted from high-fidelity simulation data serves as a fixed or lightly trainable trunk, while neural networks learn scenario-dependent modal weights or residual corrections in the branch—yielding a highly compact, flexible surrogate operator model applicable in forward, inverse, and multi-fidelity settings.

1. Methodological Foundations of POD–DeepONet Surrogates

Proper orthogonal decomposition provides an optimal low-rank projection for high-dimensional field data based on the singular value decomposition (SVD) of solution snapshots. The snapshot matrix $Y \in \mathbb{R}^{n\times m}$ formed by stacking output fields from $m$ scenarios is decomposed as $Y = U\Sigma V^T$; the leading $r$ left singular vectors $U_r = [u_1,\ldots,u_r]$ define the spatial POD basis. The field $y$ for a new scenario is then approximated by $y \approx U_r\alpha$, with $\alpha$ obtainable via projection or learned mapping.

DeepONet approximates nonlinear operators by factorizing the output function as $$\mathcal{G}(f)(x) \approx \sum_{i=1}^p b_i(f)\, t_i(x)$$, with $b_i$ learned in the branch net and $t_i$ in the trunk net. In a POD–DeepONet, the trunk is fixed to the modes $u_i(x)$ from POD, and the neural branch predicts coefficients $\alpha_i$ given the scenario parameters, design variables, or input field samples. Variants include trainable trunks (learning corrections to the POD basis) and pre-processing networks (as in flexDeepONet) that align data in latent coordinates prior to projection.

Universal operator approximation properties hold for this architecture, with global surrogate error split as $\epsilon_{\rm POD} + \epsilon_{\rm NN}$—the sum of projection error from POD truncation and network representation error in the branch projection (Sharma et al., 2024, Venturi et al., 2022, Wang et al., 1 Jan 2026).

2. Workflow: Data Preparation, SVD Compression, and Network Training

The practical POD–DeepONet surrogate workflow consists of:

• Snapshot collection: Assemble input–output pairs from high-fidelity simulations (e.g., parametric PDE solutions, PIC charge-density fields, photonic crystal band diagrams, or poroelasticity responses).
• POD basis extraction: Compute SVD or eigendecomposition of the output snapshot matrix, truncate to retain leading $r$ modes based on energy thresholds (e.g., 99%+ retained variance).
• Branch network design: Input is scenario-defining parameters (such as physical parameters, pixel layouts, reduced inputs from K–L or input-POD), output is modal weights $\alpha_i(f)$.
• Trunk definition: Use precomputed POD modes as basis/trunk; trainable trunks may also be deployed for corrections or improved generalization.
• Loss function: Typically mean squared error (MSE) or relative $\ell_2$ error on outputs, with regularization (e.g., weight decay) and scenario/field-specific scaling.
• Optimization: Adam, AdamW, or L-BFGS with decaying learning rate. Training is fast due to low-dimensional outputs (modal coefficients).

The approach admits considerable modularity; for example, the use of input and output POD (input dimensionality reduction and output mode compression, respectively) and decoupled two-step trunk/branch training (Park et al., 15 Sep 2025).

3. Architectural Variants and Extensions

Multiple architectural variants are described in the literature:

• Fixed POD trunk: Modal basis is frozen, only the branch predicts coefficients (Venturi et al., 2022, Sharma et al., 2024).
• Trainable POD trunk: Small neural trunk net learns corrections to basis (Venturi et al., 2022).
• Ensemble and MoE trunks: Mixture-of-experts or ensemble DeepONets enhance expressivity by adding local POD modes, vanilla trunks, or partition-of-unity (PoU) MoE trunk components (Sharma et al., 2024); this is crucial for regions with steep gradients or localized features.
• Pre-transformation/flexDeepONet: Pre-net learns alignment parameters (shift/rotation/scale) to place scenarios in a moving frame and reduce required mode count (Venturi et al., 2022).
• Multi-fidelity residual learning: High-fidelity data is used both to build POD basis and to learn residual correction via a branch–trunk DeepONet (Demo et al., 2023).

In photonic crystal and poroelastic inverse design, end-to-end gradient-based optimization is enabled by the differentiability of the POD–DeepONet surrogate—all operations, including binarity regularization and non-uniqueness mitigation, are efficiently addressable via backprop (Wang et al., 1 Jan 2026).

4. Quantitative Results and Benchmark Comparisons

Empirical performance of POD–DeepONet surrogates is characterized by substantial error reduction over classical model reduction and efficiency gains over fully learned or simulation-based approaches:

Application	Surrogate Error	Speedup/Benefit	Reference
Algebraic + NSE residuals	Rel. error down %%%%17$y$18%%%% over POD	No add'l HF solves; $L^2$ error $<0.01$	(Demo et al., 2023)
PIC/Poisson solving	L2 error $<$3.5% (best 0.82%)	Up to 300$\times$ Poisson speedup	(Lv et al., 27 Apr 2025)
Photonic crystal bands	Uniform surrogate error = $\epsilon_{\rm POD}+\epsilon_{\rm NN}$	Gradient-based inverse design	(Wang et al., 1 Jan 2026)
Poroelasticity	RMSE $4\times10^{-3}$ to $6\times10^{-2}$	Inference $<0.05$s/sample	(Park et al., 15 Sep 2025)
Operator learning (Darcy, Navier–Stokes, reaction–diffusion)	POD–DeepONet + MoE: $<$0.2% (2–4× vanilla)	Universal approximator, modular	(Sharma et al., 2024)

These results demonstrate robust error control and significant computational savings, with flexibility for both forward surrogacy and inverse solution generation. Addition of nonlinear corrections or local expert trunks further reduces errors in high-gradient regions.

5. Practical Implementation and Limitations

POD–DeepONet surrogates are implemented by precomputing the POD basis (no backprop required for trunks), training branch networks with parameter-efficient architectures, and tuning hyperparameters such as learning rate, batch size, and regularization. Output scaling and learning-rate annealing are used for stable training (Sharma et al., 2024, Venturi et al., 2022).

Limitations arise for physical systems with strong transport, localized features, or symmetry-induced inefficiencies in global modal decompositions. Combining global POD trunks with PoU mixture-of-experts, flexDeepONet pre-transforms, or ensemble architectures mitigates these issues. In all cases, the universal approximation theorem assures that increasing mode count and branch capacity can reduce surrogate errors arbitrarily.

6. Applications Across Domains

POD–DeepONet surrogates have been successfully deployed for:

• Multi-fidelity residual correction in reduced-order modeling, including parametric PDEs (Navier–Stokes) and algebraic test functions (Demo et al., 2023).
• Surrogate Poisson solvers embedded in kinetic plasma PIC codes, with substantial reductions in simulation time and robustness to discrete charge deposition (Lv et al., 27 Apr 2025).
• Forward and inverse design of 2D photonic crystals, coupling high-fidelity finite-element band structure computations to differentiable surrogates and inverse optimization pipelines (Wang et al., 1 Jan 2026).
• Surrogate modeling in poroelasticity for uncertainty quantification, compressing both input (K–L/POD) and output spaces and supporting efficient two-step training (Park et al., 15 Sep 2025).
• General operator learning, including fluid flows, reaction–diffusion systems, and chemical reaction dynamics, with ensemble approaches providing accuracy and interpretability (Sharma et al., 2024, Venturi et al., 2022).

The architecture supports non-intrusive, physics-informed learning and direct application in large-scale design and uncertainty quantification.

7. Future Perspectives and Generalizations

POD–DeepONet surrogates are a foundational tool in scientific operator learning. Further generalizations—such as inclusion of unstructured data, multifidelity knowledge, ensemble and local basis enrichment, and causality-aware recursive propagators (e.g., DeepPropNet (Liu et al., 2022))—are active areas of development. Hybridization with kernel-based input reduction and physics-informed loss functions further enhances stability and generalization (Park et al., 15 Sep 2025). The flexible modular construction admits integration into inverse design, surrogate-based optimization, and real-time field reconstruction pipelines. The underlying projection-based optimality of POD, combined with end-to-end neural operator training, continues to drive new efficiencies and flexibility in data-driven scientific modeling.