Spectral Normalized GP for SPINNs

• The paper introduces a spectral normalized GP module embedded within SPINNs to provide Bayesian uncertainty quantification without the cost of explicit spectral norm enforcement.
• Quantum orthogonal neural layers replace traditional spectral normalization, ensuring 1-Lipschitz continuity and scalability through inherent orthogonality.
• Numerical results demonstrate significant improvements in accuracy and calibration for high-dimensional PDE problems compared to classical methods.

A Spectral Normalized Gaussian Process (SNGP) for Separable Physics-Informed Neural Networks (SPINNs) is a recently introduced approach for embedding Bayesian uncertainty quantification into deep learning surrogates for Partial Differential Equations (PDEs). The QO-SPINN framework integrates quantum orthogonal neural layers with a SNGP module, leveraging quantum computing to achieve both computational efficiency and provable regularity. This methodology provides distance-aware uncertainty quantification with inherent Lipschitz guarantees and removes the need for explicit spectral normalization, enabling efficient and theoretically grounded application to high-dimensional PDE problems (Zanotta et al., 16 Nov 2025).

1. SPINN Architecture and Separable Ansatz

The SPINN paradigm approximates scalar PDE solutions by decomposing the network into a set of $K$ one-dimensional subnetworks and combining their outputs by a rank-$r$ canonical polyadic (CP) decomposition. For a $K$-dimensional problem, the solution representation is: $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$ where each $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$ is an MLP subnetwork. For $K=2$, this reduces to: $$u(x_1,x_2) = \sum_{i=1}^r \phi_i(x_1)\,\psi_i(x_2).$$ In QO-SPINNs, each subnetwork is realized as a Quantum Orthogonal MLP, ensuring orthogonality of the linear weights. The overall CP rank $r$ governs expressivity and computational complexity, while separation across input dimensions facilitates scalable collocation ($O(N \times K)$ vs $O(N^K)$ for dense PINN collocation).

2. Spectral Normalized Gaussian Process Layer

Within the SNGP framework, the conventional final dense layer of the (residual) SPINN is replaced by a Gaussian Process prior $r$0. Here, the covariance $r$1 is constructed over latent representations $r$2 with a kernel $r$3, typically chosen as the RBF: $r$4 By Bochner’s theorem, this allows for spectral (random Fourier) feature approximations: $r$5 with $r$6 sampled from $r$7 and $r$8 uniform over $r$9. The kernel matrix is approximated as $K$0 with $K$1.

3. Spectral Norm Enforcement: Classical and Quantum Orthogonal Layers

In standard SNGP, spectral norm constraints ($K$2) on all hidden layers are necessary to guarantee 1-Lipschitzness, enforced via power iteration or SVD, incurring $K$3 or $K$4 cost, respectively. This normalization step ensures stability and well-calibrated uncertainty.

QO-SPINNs replace each linear operation with networks of Hamming weight-preserving quantum gates (notably, real beam splitter (RBS) circuits), yielding strictly orthogonal weight matrices $K$5 with

$K$6

This ensures spectral norm regularization is inherent and cost-free, removing the bottleneck of explicit normalization and directly supporting distance awareness for the SNGP posterior.

4. Bayesian Inference and Predictive Uncertainty

The final SNGP module admits closed-form Bayesian linear regression over the projected random features. The posterior covariance on the GP regression weights $K$7 is: $K$8 The predictive mean and variance at a test feature point $K$9 are: $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$0

$$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$1

representing distance-aware, pointwise posterior uncertainty for the predicted PDE solution. This approach provides a built-in, theoretically motivated calibration tool absent in standard PINNs or MC Dropout methods.

5. Training, Losses, and Optimization

The training objective aggregates three terms:

• PDE residual loss: $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$2 (using forward-mode automatic differentiation)
• Boundary/data loss: $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$3
• GP posterior (negative log marginal likelihood): $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$4

The total loss is $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$5. Both SPINN and SNGP parameters ($$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$6) are trained jointly using gradient descent (Adam optimizer). Kernel hyperparameters (e.g., $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$7) can be tuned by maximizing the marginal likelihood or via cross-validation.

Quantum acceleration arises through quantum tomography of the orthogonal layers: the forward pass in each $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$8-dimensional subnet involves $$u(x_1,\dots,x_K) = \sum_{i=1}^r \prod_{j=1}^K \phi_{i,j}(x_j),$$9 complexity versus $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$0 for classical dense layers.

6. Computational Complexity and Scaling

The following table summarizes the key computational aspects, comparing classical and quantum SNGP for SPINNs:

Architecture	Forward Pass Complexity	Spectral Norm Enforcement
Classical SNGP	$\phi_{i,j}:\mathbb{R}\to\mathbb{R}$1	$\phi_{i,j}:\mathbb{R}\to\mathbb{R}$2 (power iteration)
QO-SPINN SNGP	$\phi_{i,j}:\mathbb{R}\to\mathbb{R}$3	None (built-in by orthogonality)

Weight update remains $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$4 in all settings. Collocation scaling for SPINN is $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$5, substantially improved over $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$6 in non-separable PINNs. This demonstrates the scalability and efficiency delivered by the quantum orthogonal architecture.

7. Uncertainty Quantification and Numerical Results

The SNGP posterior provides access to a pointwise standard deviation $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$7, reflecting the epistemic uncertainty of the model. To assess calibration, the Error-Aware Correlation (EAC) metric is used: $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$8 where $\phi_{i,j}:\mathbb{R}\to\mathbb{R}$9 is the absolute error and $K=2$0 is the predicted uncertainty. Higher EAC values indicate concordance between model error and predicted uncertainty.

Numerical benchmarks on the forward and inverse solution of PDEs provided the following highlights:

• 2D advection–diffusion: QO-SPINN MSE $K=2$1 vs. SPINN $K=2$2.
• 1D Burgers (at $K=2$3): QO-SPINN MSE $K=2$4, EAC $K=2$5 versus MC Dropout EAC $K=2$6.
• 3D advection–diffusion: QO-SPINN MSE $K=2$7 vs. SPINN $K=2$8.

These results indicate an order-of-magnitude improvement in accuracy and calibration using QO-SPINN SNGP over both classical SPINN and MC Dropout methods in tested regimes.

In summary, embedding spectral normalized Gaussian process layers in quantum-orthogonal SPINN architectures yields a framework that is provably 1-Lipschitz, avoids costly spectral normalization, supports theoretically grounded uncertainty quantification, and achieves quantum-accelerated scaling for PDE surrogate modeling (Zanotta et al., 16 Nov 2025).