Physics-Informed Learning of Probabilistic Gegenbauer Reconstruction for Transport-Dominated Problems

Abstract: Transport-dominated problems remain challenging for data-driven methods, which often exhibit severe numerical oscillations near shocks or steep gradients due to globally supported basis functions or overly smooth hypothesis spaces. Gegenbauer reconstruction has shown promise in mitigating such oscillations, but its effectiveness critically depends on the reconstruction parameters, particularly the weight parameter $λ$ and truncation order $m$. For data-driven models, variations in governing problems, training data, and model architectures make systematic parameter selection particularly challenging. To address this issue, we propose a physics-informed machine-learning framework that predicts probability distributions over candidate Gegenbauer parameter pairs, enabling probabilistically weighted reconstruction while accounting for parameter uncertainty. A two-stage strategy is adopted, in which a general predictor is first pre-trained and then fine-tuned for target problems to balance accuracy and computational cost. The framework is evaluated for reduced-order and neural operator models, represented by POD-Galerkin and DeepONet, respectively. Numerical experiments on one- and two-dimensional transport-dominated problems show that the framework learns effective spatially adaptive parameter distributions. Compared with conventional reconstruction strategies, it reduces numerical errors by up to one to two orders of magnitude and achieves a more favorable accuracy--cost trade-off than problem-specific model retraining.

• The paper introduces a CNN that predicts probability distributions over Gegenbauer reconstruction parameters, reducing mean relative errors by roughly 2.6–12× across five one- and two-dimensional transport benchmarks.
• The method combines entropy-regularized Gibbs targets, physics-guided reconstruction losses, and two-stage pre-training with target-specific fine-tuning, achieving near-retraining accuracy using about 100–250 samples instead of more than 2,000.
• The results show that fine-tuning mainly the prediction head is effective and inexpensive, but performance depends on accurate discontinuity detection, problem-specific adaptation, and improved handling of input noise and online parameter-sweep costs.

Motivation and problem setting

Data-driven solvers for transport-dominated problems—reduced-order models built on global POD bases and neural operators such as DeepONet—routinely produce spurious oscillations near shocks and steep gradients. The paper addresses a specific bottleneck in a post-processing remedy: Gegenbauer reconstruction, originally introduced to suppress the Gibbs phenomenon in spectral approximations [gottlieb1997gibbs], was recently extended to reduced-order modeling by the authors [yan2026gegenbauer], but its performance depends sensitively on the weight parameter $\lambda$ and truncation order $m$. In classical spectral settings, theoretical guidelines exist for choosing $(m,\lambda)$ relative to the truncation level $N$; in data-driven settings no such structure exists, because the effective approximation space is shaped implicitly by the problem, training data, architecture, and optimization process. Moreover, the optimal parameters are spatially varying. The contribution is a physics-informed machine-learning framework that predicts an input-dependent probability distribution over candidate parameter pairs $(m,\lambda)$, from which a weighted reconstruction is formed.

Probabilistic parameter selection

The core formulation relaxes deterministic minimum-error selection over the parameter grid $\Gamma = \mathcal{M}\times\Lambda$ into an entropy-regularized problem on the probability simplex. Given reconstruction errors $E_{i,j}$ against a reference solution, the authors minimize the expected error plus a Shannon entropy penalty, yielding a strictly convex problem whose unique solution is the Gibbs distribution

$$P_{\gamma,i,j}^\star \propto \exp(-\gamma E_{i,j}).$$

This construction has two useful properties. First, it provides a variational justification for converting the error matrix into soft targets: as $\gamma\to\infty$ the distribution collapses to the one-hot minimizer, while finite $\gamma$ retains information from multiple near-optimal candidates—an important robustness mechanism since several parameter pairs often yield comparable errors. Second, the resulting weighted reconstruction $m$0 directly embeds parameter uncertainty into the post-processing output. The authors are careful to note that $m$1 is not a Bayesian posterior but an error-induced soft representation of selection uncertainty; this is an honest framing, since no probabilistic semantics beyond the weighting role are claimed.

The predictor itself is a multi-scale CNN (CNN_ReconParamNet, roughly 216k parameters) with parallel convolutional branches of kernel sizes 3, 5, and 7, fused and mapped through a fully connected head to logits over $m$2. Training combines a reconstruction fidelity loss on the probability-weighted output with a KL divergence term matching the predicted distribution to the Gibbs soft target, weighted by a mixing coefficient $m$3. This objective is described as physics-guided in that it exploits the analytic structure of Gegenbauer reconstruction and reference-solution data rather than fitting parameters heuristically.

Supporting components: oscillation discrimination and two-stage training

Because reconstruction proceeds interval-by-interval between detected discontinuities, the framework requires classifying oscillations along each one-dimensional cut as either discontinuity-induced (DIOs) or spurious oscillations in smooth regions (SOSs). The proposed criterion uses a Sobel-type gradient operator to locate candidate discontinuities, then computes the energy-weighted centroid of bandpass-filtered high-frequency content together with an edge-to-interior energy ratio $m$4. Oscillations with energy concentrated near interval boundaries ($m$5, or normalized centroid within an edge fraction) are classified as DIOs. The thresholds ($m$6–$m$7, edge fraction $m$8–$m$9) are empirical, and the criterion is explicitly presented as a problem-dependent working definition rather than a rigorous classifier—a limitation that propagates into dataset labeling.

Training follows a pre-training/fine-tuning paradigm. Pre-training uses 2000 low-cost samples per oscillation class: DIO samples are generated by projecting shifted smooth profiles onto a precomputed reduced space (avoiding repeated full solves), and SOS samples by perturbing smooth baselines with localized sinusoidal and filtered stochastic components. Fine-tuning then adapts only the final prediction head on a small target-specific dataset (roughly 100–250 samples), freezing the convolutional backbone.

Numerical results

The framework is evaluated on four benchmarks—1D linear transport, 1D inviscid Burgers (with both G-ROM and DeepONet baselines), 2D linear transport, and 2D Burgers—with G-ROM solutions as inputs and line-by-line reconstruction in 2D. Errors are reported on masked grids excluding five points around each detected discontinuity.

The headline quantitative findings are substantial:

Benchmark	Baseline error	Fine-tuned error	Reduction
1D linear (mean rel. $(m,\lambda)$0, 50 samples)	$(m,\lambda)$1	$(m,\lambda)$2	~7×
1D Burgers G-ROM (mean rel. $(m,\lambda)$3, 52 samples)	$(m,\lambda)$4	$(m,\lambda)$5	~12×
1D Burgers DeepONet (mean rel. $(m,\lambda)$6, 55 samples)	$(m,\lambda)$7	$(m,\lambda)$8	~2.6×
2D linear (mean rel. $(m,\lambda)$9, 38 samples)	$N$0	$N$1	~5×
2D Burgers (mean rel. $N$2, 44 samples)	$N$3	$N$4	~10×

Fine-tuned models consistently match or nearly match fully re-trained models—for example, mean relative $N$5 errors of $N$6 versus $N$7 on the 1D linear problem—at a fraction of the cost: fine-tuning used roughly 100–230 samples and under 12 s of training versus roughly 2000+ samples and 280–380 s for re-training. In the 2D cases, fine-tuning reduces dataset generation and training costs by more than an order of magnitude. A comparison against total variation regularization on the 2D linear problem shows TV only partially suppresses oscillations, supporting the advantage of the Gegenbauer-based approach.

Two caveats temper these results. First, the pre-trained model alone performs poorly on several benchmarks—it can even degrade the raw solution (e.g., mean relative $N$8 of $N$9 versus $(m,\lambda)$0 for raw DeepONet predictions)—so problem-specific adaptation is not optional but essential, particularly in nonlinear and 2D settings. Second, the dominant online cost in 2D comes from evaluating reconstructions over all candidate pairs (~47.5 s per sample), so the learned predictor improves accuracy but not online evaluation speed.

Fine-tuning analysis

Three ablations clarify the adaptation behavior. With respect to dataset volume, reconstruction errors decrease rapidly with small amounts of target data and then follow approximately the Monte Carlo rate $(m,\lambda)$1, suggesting the remaining error is sampling-limited rather than optimization-limited. With respect to trainable components, updating only the final fully connected head (19.62% of parameters) achieves accuracy comparable to full fine-tuning, whereas updating only convolutional branches performs markedly worse (validation loss $(m,\lambda)$2 versus $(m,\lambda)$3 for head-only). This supports the interpretation that the pre-trained backbone learns transferable oscillation-aware features and that adaptation mainly recalibrates the feature-to-distribution map. Out-of-distribution tests show consistent improvement under temporal shifts (errors again tracking the Monte Carlo trend), but noise-robustness saturates above the Monte Carlo reference, indicating that robustness to input perturbations is not governed by fine-tuning data volume alone and may require explicit denoising mechanisms.

Limitations and open questions

Several limitations are acknowledged or evident. The oscillation-discrimination thresholds are hand-tuned and problem-dependent, and misclassification would corrupt both training labels and interval partitioning. The entropy temperature $(m,\lambda)$4 and loss mixing weight $(m,\lambda)$5 are fixed hyperparameters without systematic selection guidance. The evaluation is restricted to scalar transport problems with parametric initial conditions on uniform grids; extension to systems, non-periodic boundaries, and more complex multiscale PDEs remains untested. The noise-OoD saturation leaves open whether robustness-aware training or denoising stages are needed. Finally, the online cost of sweeping the full parameter grid motivates cheaper candidate-selection strategies, which the paper does not address.

Conclusion

The paper formulates Gegenbauer parameter selection as entropy-regularized probabilistic inference, trains a multi-scale CNN to predict the resulting distributions, and demonstrates that probability-weighted reconstruction reduces errors by up to one to two orders of magnitude across ROM and neural-operator baselines in 1D and 2D transport problems. The two-stage strategy achieves re-training-level accuracy at an order-of-magnitude lower offline cost, with head-only fine-tuning sufficient in practice. The method's dependence on accurate discontinuity detection, its empirical hyperparameters, and its demonstrated sensitivity to input noise remain the principal open issues.