POD–DeepONet Surrogate for Operator Learning

• The paper introduces POD–DeepONet surrogate, combining a fixed POD trunk with a neural network branch to decouple spatial variation from parameter dependence.
• Experimental results in PDEs, photonic crystal design, and PIC simulations demonstrate error reductions of 0.3%-0.99% and significant parameter count savings.
• The approach offers explicit error control, efficient parallel training, and enhanced interpretability, making it ideal for complex, high-dimensional operator tasks.

A POD–DeepONet surrogate is a reduced-order operator learning framework that combines Proper Orthogonal Decomposition (POD) with the Deep Operator Network (DeepONet) architecture to produce highly efficient, physically interpretable surrogates for high-dimensional and nonlinear operators, especially in settings dominated by multiscale PDEs, structured scientific data, or particle simulations. The approach leverages data-driven low-rank decomposition of operator outputs (POD) to establish a fixed, compact basis (the trunk), while a neural network learns to map system inputs onto the reduced coefficient space (the branch). This decomposition decouples spatial variation from parameter/function dependence and provides explicit error control, efficiency, and differentiability.

1. Vanilla DeepONet and the Rationale for POD Integration

DeepONet approximates a nonlinear operator $\mathcal{G}: u(\cdot)\mapsto v(\cdot)$ by projecting input samples $u$ (taken at fixed sensor locations) onto nonlinear features via a branch network $\boldsymbol{b}(u)$, and projecting the output coordinate $y$ onto nonlinear features via a trunk network $\boldsymbol{t}(y)$. The operator output is approximated as a sum of products,

$$\widehat{\mathcal{G}(u)(y)} = \boldsymbol{b}(u)\cdot \boldsymbol{t}(y) + b_0,$$

where $b_0$ is an optional bias term. This architecture is universal for operator learning but can require substantial network capacity to achieve high fidelity for complex systems or where solution manifolds are low-dimensional but highly structured (Venturi et al., 2022).

POD–DeepONet replaces the trainable trunk with a fixed basis from POD, compressing the solution space to its most energetic modes. This ensures interpretability, reduces trainable parameters, and enables parallel training, while providing an explicit source of approximation error (POD truncation). All nonlinear parameter dependence is offloaded to the branch network, which maps inputs $u$ to modal coefficients.

2. POD–DeepONet Architecture and Workflow

The POD–DeepONet surrogate shares the following workflow across implementations (Sharma et al., 2024, Wang et al., 1 Jan 2026, Lv et al., 27 Apr 2025):

1. Snapshot collection: Generate a library of solution samples $v_i$ of the operator (e.g., solution fields, band structures, potentials) across varying initial conditions, parameters, or design vectors.
2. POD basis construction: Assemble the snapshot matrix, standardize (zero mean, unit variance), and extract the first $r$ left singular vectors (POD modes) $\{\phi_k\}_{k=0}^{r-1}$ via SVD or eigendecomposition of the snapshot correlation. Mode truncation $r$ is chosen to achieve target energy capture.
3. Frozen trunk: The set of $r$ POD modes forms the trunk, fixed throughout network training, and provides a physically meaningful, low-dimensional basis for the output function $v(y)$.
4. Branch network: A feed-forward network is trained to map the input function or parameterization (e.g., PDE coefficient, design variables, charge density coefficients, ...) to the $r$ modal coefficients $\beta(u)$.
5. Surrogate evaluation: For test input $u$, the branch produces $\beta(u)\in \mathbb{R}^r$; the output is reconstructed as $\hat v(y)=\sum_{k=0}^{r-1}\beta_k(u)\phi_k(y)$.

This separation enables efficient end-to-end training even with large $r$ and provides a modular, interpretable surrogate that preserves physical structure and modal energy (Sharma et al., 2024).

3. Applications and Empirical Performance

3.1 Scientific PDE Operators

For parametric or random-coefficient PDEs (e.g., Darcy flow, cavity flow, reaction-diffusion), POD–DeepONet produces surrogates with relative $\ell_2$ test errors of $0.3\%$–$0.33\%$ (Darcy, cavity) and MSE as low as $5\times10^{-7}$ (reaction-diffusion), using typical branch networks of 3–5 layers with 64–128 neurons and $r=10$–$40$ modes (Sharma et al., 2024).

3.2 Photonic Crystal Design

In 2D photonic crystal band-structure mapping and inverse design, the POD–DeepONet surrogate uses a POD trunk (rank $r=25$) extracted from high-fidelity band snapshots. The branch MLP (2–3 hidden layers, width 128–256) maps binary pixel design vectors (dimension $N_f=36$ due to symmetry) to the low-dimensional band representation. Test-set mean relative error is $0.46\%$ for the forward map, and $0.99\%$ for inverse dispersion-to-structure design. Bandgap inverse design achieves gap-edge MAEs on the order of $0.008$–$0.009$, and band width rMAE $0.0578$ (outperforming full MLP baselines) (Wang et al., 1 Jan 2026).

3.3 Particle-in-Cell Simulation Surrogates

PaRO-DeepONet partitions charge density snapshots from multi-run PIC simulations by particle sparsity, performs local POD on each partition, and concatenates modal coefficients as input to the branch network. Across four plasma benchmark cases, the error in potential field prediction using POD–DeepONet ranges from $0.0102$ to $0.032$ (relative $\ell_2$), and as low as $0.0090$ with PaRO methods. Computational speed-ups of $31.8\times$ (total PIC time), and $287\times$–$300\times$ (Poisson solver) are reported, with robustness to sparse charge deposition and generalization to complex geometries (Lv et al., 27 Apr 2025).

Application area	Test error (rel. $\ell_2$)	Parameter count reduction	Additional notes
PDE (Darcy, etc.)	$\sim0.30\%$–$0.33\%$	$>90\%$	Energy-based $r$ selection
Photonic crystal design	$0.46\%$ (forward), $0.99\%$ (inverse)	$>80\%$	Differentiable, supports gradient inverse
PIC/Poisson surrogate	$0.0102$–$0.032$	$>90\%$	68–85% faster overall, 99.6% Poisson runtime

4. Extensions: Symmetry Handling and Model Flexibility

In problems characterized by continuous or discrete symmetries (translation, rotation, scaling), unmodified POD–DeepONet can require high-rank trunks to represent solution orbits. The flexDeepONet extension introduces a pre-network that maps input $u$ to symmetry parameters (e.g., shift $\bar y(u)$, rotation $\bar\theta(u)$, scale $\bar s(u)$), applies a coordinate transformation $T_\theta(y;u)$, and projects trunk modes onto this transformed frame. This reduces the intrinsic dimensionality, capturing the symmetry-free dynamics in a compact subspace. For instance, flexDeepONet achieved $99\%$ parameter reduction and substantial RMSE improvement over vanilla DeepONet for 2D rigid-body operator tasks (Venturi et al., 2022).

Limitations

• Pre-network specialization: flexDeepONet currently handles only rigid (affine) symmetries; nonrigid or manifold deformations would require more general (e.g., autoencoder-based) latent transforms.
• POD–DeepONet’s SVD step requires grid-aligned snapshots; vanilla DeepONet allows for unstructured data.
• POD truncation risk: too few modes induce bias, too many with small datasets risk overfitting.

5. Training Paradigms, Error Metrics, and Best Practices

Training Strategies

• The branch network can be trained via coefficient regression (matching true modal coefficients) or end-to-end minimization of mean squared error between reconstructed field and reference.
• The POD trunk remains fixed post-decomposition, yielding a significant reduction in trainable parameters and decoupling training (Sharma et al., 2024).
• Adam or AdamW are the standard optimizers, with learning rates in $10^{-3}$–$10^{-4}$ and weight decay for regularization.

Error Characterization

The total surrogate error decomposes as

$$\|W_h(d)-\widehat W_h(d)\|_F \leq \underbrace{\|W_h(d)-\Pi_r W_h(d)\|_F}_{\text{POD truncation}} + \underbrace{\|\Pi_r W_h(d) - \widehat W_h(d)\|_F}_{\text{Network error}},$$

making the separation between spatial/temporal approximation and parametric regression explicit (Wang et al., 1 Jan 2026).

Best Practices

• The mode number $r$ should capture at least $99\%$ of the energy spectrum to avoid severe truncation bias.
• Input and output normalization is crucial for stable training and generalization.
• Validation should include monitoring for overfitting (training vs. validation error), visualization of reconstructed fields for high-mode artifacts, and mode-energy inspection via SVD.
• Early stopping or cross-validation is advised due to the risk of overfitting with large $r$ and small training sets.

6. Theoretical Properties and Future Directions

POD–DeepONet surrogates inherit continuity and universal approximation properties under mild assumptions. For any fixed POD trunk, a sufficiently wide branch network can achieve arbitrarily small network error on the truncated subspace (Wang et al., 1 Jan 2026). The mapping remains differentiable (almost everywhere) in the design or function input space, supporting end-to-end differentiable inverse design and constrained optimization.

Open research directions include:

• Robust POD/PCA methods for outlier-resilient mode discovery (Venturi et al., 2022).
• Incremental and tensor decompositions to handle streaming or multi-physics multi-input operator data, moving beyond static SVD (Venturi et al., 2022).
• Hybrid DeepONet-autoencoder architectures for non-affine symmetry and manifold learning.
• Theory: Explicit error bounds connecting operator smoothness, POD spectrum decay, and network approximation rate (Venturi et al., 2022).
• Full end-to-end integration with physical simulation codes, especially for nonlinear or stochastic multiphysics (Lv et al., 27 Apr 2025).

7. Summary and Significance

POD–DeepONet surrogates provide a robust, interpretable, and computationally efficient framework for operator learning in high-dimensional or physics-based settings. By embedding the dominant POD modes as a fixed trunk and leveraging neural networks as parameter-to-coefficient regressors, these surrogates achieve orders-of-magnitude parameter savings, explicit error control, and competitive accuracy across diverse domains, including parametric PDEs, photonic device design, and plasma simulation. The approach is actively under investigation for further extensions in symmetry handling, multi-fidelity coupling, and theoretical error characterization (Venturi et al., 2022, Sharma et al., 2024, Wang et al., 1 Jan 2026, Lv et al., 27 Apr 2025).