LowNet: Looped Wavelet Neural Networks with 24 Learnable Parameters for Solving Poisson Equations

Yuanhao Gong

cs.AIOct 18, 2026

LowNet: Looped Wavelet Neural Networks with 24 Learnable Parameters for Solving Poisson Equations

Authors: Yuanhao Gong

Organizations: Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences

Abstract

Poisson equations frequently appear in various fields, such as mathematics, physics and informatics. Solving Poisson equation, however, is challenging. Traditional methods are not data-specific, although have high certainty. They require a large number of iterations due to the CFL numerical stability. In contrast, deep learning methods are data-driven and more efficient, but require accurate training data and suffer from heavy computation. To tackle these issues, we propose a wavelet-inspired Poisson solver with 24 trainable scalars. The proposed network has several advantages. First, it is compact. It has four independent $5\times5$ kernels, each represented by six parameters, leading to 24 scalars in total. Such compact format can be deployed on most devices. Second, thanks to the loop design, it achieves higher accuracy than previous methods. A residual loop reuses the wavelet block without increasing its parameters. Numerical experiments confirm that such loop indeed improves the solving accuracy. On 10000 images, one, two, and three loops of the wavelet block lead to PSNRs of $47.27$ , $74.14$ , and $100.74$ dB, respectively. Third, the proposed method is computationally fast. Thanks to the wavelet transform, the proposed method has linear computation complexity, leading to high computation performance. In short, we propose a novel looped wavelet neural network with compactness, accuracy and efficiency for solving Poisson equations. Our method can be applied in various fields, such as computational fluid dynamics, image reconstruction, biomedical engineering, electromagnetic field simulation.

Figures & tables

Fig. 1: The learnable kernels $H$ , $G$ , $K$ are shared across color channels and scales. The proposed network is compact. There are 24 scalar parameters in total. Moreover, the wavelet is looped to improve the accuracy. Finally, this network has linear computation complexity.

Panel of Figure 2 — Fig. 2: The wavelet block and its three-prefix training unroll. In (a), each stored field has three independently processed colors with shared coefficients. The physical one-pixel zero frame and computational five-pixel halo are distinct. The four radial banks remain independent, while $H/G/K$ are reused across the seven fine levels and $G_{c}$ is used only at the deepest level. In (b), every occurrence of $\mathcal{B}_{\theta}$ uses the same 24 parameters. The operator $\Delta$ produces the complete zero-framed Laplacian and $M$ the framed-image mean. The decoder internally uses $L$ ; the physics loss instead forms the interior Dirichlet residual.

$R$	Original177	Radial24(supervised)	LowNet24(unsupervised)
1	$43.322\pm 0.001$	$47.273\pm{<}0.001$	$47.272\pm{<}0.001$
2	$61.958\pm 0.005$	$74.123\pm 0.008$	$74.136\pm 0.008$
3	$76.585\pm 0.016$	$100.727\pm 0.017$	$100.741\pm 0.017$
4	$89.406\pm 0.035$	$119.909\pm 0.012$	$119.924\pm 0.006$
5	$100.835\pm 0.060$	$121.375\pm 0.008$	$121.378\pm 0.012$
6	$110.738\pm 0.076$	$121.399\pm 0.018$	$121.403\pm 0.008$

TABLE I: Clean test mean per-image PSNR (dB), mean

\pm

three-fit sample SD. Radial24 is supervised with a single-output pixel loss; LowNet24 uses the connected R1–R3 RHS spectral loss. All fixed R1–12 points are retained. A reported

\pm{<}0.001

denotes nonzero sample SD below 0.001 dB, not zero uncertainty. SD is not a scene-level confidence interval.

Fig. 3: Frozen-weight clean reconstruction for all measured call counts. Curves use mean per-image PSNR and mean pixel MSE on the same ordered 10,008 test patches. The exact CPU FP64 DST-I reference uses the same FP32 complete Laplacian and is a numerical accuracy reference; a matched GPU runtime was not measured. Fit SD is small relative to the plotted range.

Kernels	$P$	H/G/K/Gc	$R=1$	$R=2$	$R=3$
free68	68	$5/3/5/3$	$43.570\pm 0.001$	$62.305\pm 0.005$	$77.270\pm 0.006$
radial18	18	$5/3/5/3$	$43.501\pm 0.001$	$62.176\pm 0.002$	$77.214\pm 0.007$
free100	100	$5/5/5/5$	$47.496\pm{<}0.001$	$74.289\pm 0.005$	$101.023\pm 0.023$
radial24	24	$5/5/5/5$	$47.273\pm{<}0.001$	$74.123\pm 0.008$	$100.727\pm 0.017$

TABLE II: Supervised single-output 100-epoch controls: free dense kernels and shell-tied kernels at mixed versus full

5\times 5

supports. These earlier metrics were scored with FP32 error arithmetic; they are not pooled with the FP64-scored core results. SDs below 0.001 dB are shown as

\pm{<}0.001

Fig. 4: Earlier supervised single-output kernel-size and dense-width controls. Panels vary the coarse detail support, fine-kernel supports, and dense

C_{f}\times C_{f}

feature width. These dense-width controls differ from the independent 24/48/72-parameter paths in Table IV ; the experiment families are not pooled.

Model	$R$	$P$	PSNR(dB)	MSE 255	Batch1(ms)	Batch128( $\mu$ s/img)
untied24	1	24	$47.272\pm{<}0.001$	1.422	0.430160	10.152
untied24	2	24	$74.136\pm 0.008$	0.002929	0.845984	19.944
untied24	3	24	$100.741\pm 0.017$	6.448e-06	1.247232	30.200
tied18	1	18	$46.340\pm{<}0.001$	1.774	0.431104	10.020
tied18	2	18	$72.848\pm 0.007$	0.004086	0.842560	19.944
tied18	3	18	$98.316\pm 0.020$	1.226e-05	1.248256	30.272

TABLE III: H=K reduces 24 to 18 parameters with identical convolution geometry. Accuracy is three-fit mean/SD; GPU times use seed 08 with 150 events per condition on the same idle GPU. Initial moment recovery and residual encoding are included. The initializer discards old K, so initial functions are not matched. SDs below 0.001 dB are shown as

\pm{<}0.001

Fig. 5: Five completed random-initialization fits versus the preserved projected-initialization reference. All random runs use the fixed 100-epoch budget and retain their best epoch-100 checkpoints. The full spread and call dependence are shown; poor accuracy is not labeled numerical divergence, and continued validation descent limits conclusions about longer optimization.

Model	$P$	$\sigma_{255}$	$R=1$	$R=2$	$R=3$
baseline177	177	0	$43.322\pm 0.001$	$61.958\pm 0.005$	$76.585\pm 0.016$
baseline177	177	5	$33.637\pm 0.001$	$34.284\pm{<}0.001$	$34.165\pm{<}0.001$
baseline177	177	10	$28.060\pm 0.001$	$28.271\pm{<}0.001$	$28.144\pm{<}0.001$
baseline177	177	20	$22.160\pm 0.001$	$22.253\pm{<}0.001$	$22.124\pm{<}0.001$
unsupervised24	24	0	$47.272\pm{<}0.001$	$74.136\pm 0.008$	$100.741\pm 0.017$
unsupervised24	24	5	$33.900\pm{<}0.001$	$34.156\pm{<}0.001$	$34.151\pm{<}0.001$

TABLE IV: Clean-trained fixed-weight reconstruction with Gaussian image noise before complete Laplacian encoding. No input/output clipping or clean mean is used. Positive noise levels have three paired fixed realizations;

\pm

is three-fit sample SD after realization/patch averaging.

\pm{<}0.001

denotes a nonzero SD below 0.001 dB.

Fig. 6: Gaussian noise is added to RGB image pixels before the complete Laplacian is computed. Scores use the clean target, frozen clean-trained models, paired noise realizations, and recovered noisy means. Improving the equation solve approaches recovery of the noisy image. The noise-variance references describe inversion, not a claimed denoising bound for estimators with an image prior.

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