Organizations: National Center for Applied Mathematics Beijing, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · The Academy for Multidisciplinary Studies, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · Academy of Information Network Security, People’s Public Security University of China, No.1 Muxidi Nanli, Beijing, 100038, China · Institute of Nuclear and New Energy Technology, Tsinghua University, Haidian District, Beijing, 100084, China · School of Mathematical Sciences, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China
Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point-to-point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point-to-function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.
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Figure 1: This comparative study demonstrates two approaches for approximating y=sin(x) . (a): Explicit polynomial fitting reveals a critical trade-off: low-degree polynomials (e.g., 3rd-degree, red) fail to capture the curvature of the function within the sampled region (x∈[−1,1]) , whereas higher-degree fitting (5th-7th degrees, orange/green) provides improved in-sample accuracy at the cost of severe overfitting and unstable extrapolation (wild oscillations beyond ∣x∣>1 ). (b): Taylor series expansion exhibits progressive refinement: lower-order approximations (1-2 terms, red/brown) approximate the core trend, whereas higher-order expansions (3-4 terms, green/blue) systematically enhance both the local accuracy and global extrapolation stability of the system by aligning derivative constraints with the intrinsic physics of sin(x) . These divergent extrapolation behaviours underscore the superiority of series expansions over purely empirical polynomial regression schemes.
Figure 2: Comparison among solutions for the heat equation: (a) The PINN. (b-c) The SineGEN and GaussGEN developed under the proposed GEN framework. (d-e) The temporal profiles produced at x = 0.5 (left) and x = 1.0 (right), demonstrating increasing extrapolation errors beyond the training domain ( t>2.0 ), which is particularly evident in the PINN predictions. (f-g) The 25 basis functions learned by the two GEN methods.
Figure 3: Wave equation: (a) Heatmap comparison among the numerical solution, the PINN and the two GEN results, with the colour intensities representing the solution values. (b) and (c) Temporal snapshots of the solutions produced at spatial locations x=0.5 and x=3.0 , respectively. The green shaded area represents the fitting region (training domain), and the red shaded area represents the extrapolation region.
Figure 4: Burgers’ equation: (a)-(c) Heatmap comparisons between exact solutions and GENs with various numbers of basis functions. (d)-(f) Spatial snapshots of the solutions produced at temporal positions t=0.25,0.5 and 0.75 , respectively. Magnified views near the maxima and minima are shown in the upper-right and lower-left corners, respectively. Marked fitting accuracy discrepancies are observed when a smaller number of basis functions is employed.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.
Hilderbrand Department of Petroleum and Geosystems Engineering, The University of Texas at Austin · Department of Earth and Planetary Sciences, The University of Texas at Austin
Department of Applied Mathematics, Mathematics, Illinois Institute of Technology, Chicago, IL, 60616 · Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, 72701 · Department of Mathematics, University of Houston, Houston, TX, 77204