Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks

Tetsuro Tsuchino; Motoki Shiga

cs.LGMar 24, 2026

Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks

Authors: Tetsuro Tsuchino, Motoki Shiga

Organizations: Graduate School of Engineering, Gifu University, Japan · International Center for Synchrotron Radiation Innovation Smart, Tohoku University, Japan · Unprecedented-scale Data Analytics Center, Tohoku University, Japan · Graduate School of Information Sciences, Tohoku University, Japan · Center for Basic Research on Materials, National Institute for Materials Science, Japan · Center for Advanced Intelligence Project, RIKEN, Japan

Abstract

In solving partial differential equations (PDEs), machine learning utilizing physical laws has received considerable attention owing to advantages such as mesh-free solutions, unsupervised learning, and feasibility for solving high-dimensional problems. An effective approach is based on physics-informed neural networks (PINNs), which are based on deep neural networks known for their excellent performance in various academic and industrial applications. However, PINNs struggled with model training owing to significantly slow convergence because of a spectral bias problem. In this study, we propose a PINN-based method equipped with a coordinate-encoding layer on linear grid cells. The proposed method improves the training convergence speed by separating the local domains using grid cells. Moreover, it reduces the overall computational cost by using axis-independent linear grid cells. The method also achieves efficient and stable model training by adequately interpolating the encoded coordinates between grid points using natural cubic splines, which guarantees continuous derivative functions of the model computed for the loss functions. The results of numerical experiments demonstrate the effective performance and efficient training convergence speed of the proposed method.

Figures & tables

Figure 1: Schematic of the proposed method, coordinate encoding on linear grid (CELG).

	H-Spline	PIXEL	CELG
Space	$\mathcal{O}(S^{D}\cdot R^{D})$	$\mathcal{O}(M\cdot R^{D})$	$\mathcal{O}(M\cdot D\cdot R)$
Time	$\mathcal{O}(S^{D}\cdot 2^{D})$	$\mathcal{O}(M\cdot 2^{D})$	$\mathcal{O}(M\cdot D\cdot R)$

Table 1: Computational complexities

Figure 2: Training results utilizing linear, cosine with/without multigrid (MG), Hermite and natural interpolations for 1D Poisson PDE. (a) Predictions of

u(x)

, (b) first and (c) second derivatives. (d) Learning curves. In (a)–(c), the analytical solution is shown as a solid gray line.

Figure 3: Analytical solutions of (a) 1D and (b) 2D multi-band Poisson PDEs.

Figure 4: Prediction results for the 1D multi-band Poisson PDE during training at (a) 100, (b) 1,000, (c) 5,000, and (d) 10,000 training epochs. The grid parameter was set to

R=16

in PIXEL, H-Spline, and CELG.

Figure 5: Learning curves for the 1D multi-band Poisson PDE. The grid parameters were set to (a)

R=4

, (b) 8, (c) 16, and (d) 32.

Panel of Figure 6 — Figure 6: Predictions for the 2D multi-band Poisson PDE by non-grid-based models (Vanilla and CP-PINN) and CELG, CELG-CP ( $R=16$ ) after (a) 1,000, (b) 5,000, and (c) 50,000 training epochs.

Figure 8: Learning curves for the 2D multi-band Poisson PDE. (a) Curves of CELG and CELG-CP with

R=16

and non-grid methods (Vanilla and CP-PINN), and those of grid-based methods with (b)

R=8

, (c)

R=16

, and (d)

R=32

	Allen–Cahn	Burgers	Flow-Mixing	Helmholtz
Domain	$[0,1]$ $\times$ $[-1,1]$	$[0,1]$ $\times$ $[-1,1]$	$[0,8]$ $\times$ $[-4,4]^{2}$	$[0,1]^{3}$
#Colloc.	50 $\times$ 128	50 $\times$ 128	$64^{3}$	$64^{3}$
#IC	128	128	$64^{2}$	-
#BC	$50\times 2$	$50\times 2$	$4\times 64^{2}$	$6\times 64^{2}$
$\bm{\lambda}$	(0.1, 1, 1)	(1, 1, 0.001)	(100, 1, 1)	(1, -, 1)
#Test	$100\times 256$	$100\times 256$	$128^{3}$	$128^{3}$

Table 2: Training settings for benchmark PDEs.

Method	$R$	Allen-Cahn/Burgers	Flow-Mixing	Helmholtz
Vanilla	–	12,737	12,801	1,985
CP-PINN	–	25,216	37,824	5,808
PIXEL	8	18,753	101,697	24,433
	16	45,377	637,249	158,321
	32	147,777	4,608,321	1,151,089
H-Spline	8	576	13,824	13,824

Table 3: Number of parameters for benchmark PDEs.

Method	$R$	Allen-Cahn	Burgers	Flow-Mixing	Helmholtz
Vanilla	–	1.73 $\pm$ 0.03	1.46 $\pm$ 0.01	1.85 $\pm$ 0.01	24.73 $\pm$ 0.02
CP-PINN	–	2.04 $\pm$ 0.05	1.71 $\pm$ 0.04	1.90 $\pm$ 0.02	4.31 $\pm$ 0.23
PIXEL	8	1.68 $\pm$ 0.07	2.26 $\pm$ 0.01	34.83 $\pm$ 0.01	24.86 $\pm$ 0.01
	16	1.60 $\pm$ 0.07	2.07 $\pm$ 0.01	27.57 $\pm$ 0.01	18.77 $\pm$ 0.02
	32	1.63 $\pm$ 0.11	2.01 $\pm$ 0.01	30.69 $\pm$ 0.01	17.53 $\pm$ 0.02
H-Spline	8	0.88 $\pm$ 0.13	0.67 $\pm$ 0.04	19.54 $\pm$ 0.29	21.44 $\pm$ 0.20

Table 4: Training time (ms./iter.) for benchmark PDEs.

Figure 9: Learning curves for benchmark PDEs: (a) Allen-Cahn, (b) Burgers, (c) Flow-Mixing, and (d) Helmholtz equations.

Panel of Figure 10 — Figure 10: Prediction results for (a) Allen-Cahn, (b) Burgers, (c) Flow-Mixing, and (d) Helmholtz equations. The third rows in (a) and (b) show the prediction results and the analytical solutions (gray) at time $t=1$ . Predictions and residuals at (c) $t=8$ and (d) $x_{3}=0.75$ .

Method	$R$	Allen-Cahn	Burgers	Flow-Mixing	Helmholtz
Vanilla	–	0.46 $\pm$ 0.23	0.08 $\pm$ 0.05	0.02 $\pm$ 0.00	1.08 $\pm$ 0.13
CP-PINN	–	0.88 $\pm$ 0.11	0.09 $\pm$ 0.00	0.05 $\pm$ 0.00	0.07 $\pm$ 0.01
PIXEL	8	0.04 $\pm$ 0.01	0.09 $\pm$ 0.05	0.01 $\pm$ 0.00	0.27 $\pm$ 0.03
	16	0.62 $\pm$ 0.02	0.47 $\pm$ 0.06	0.01 $\pm$ 0.00	0.14 $\pm$ 0.00
	32	0.99 $\pm$ 0.04	0.94 $\pm$ 0.04	0.05 $\pm$ 0.01	0.68 $\pm$ 0.00
H-Spline	8	0.62 $\pm$ 0.00	0.47 $\pm$ 0.02	0.18 $\pm$ 0.00	0.54 $\pm$ 0.00

Table 5: Relative errors for benchmark PDEs.

Hyperparameter	2D	3D	4D	5D
Domain	$[0,1]^{2}$	$[0,1]^{3}$	$[0,1]^{4}$	$[0,1]^{5}$
#Colloc.	$64^{2}$	$64^{3}$	$16^{4}$	$16^{5}$
#IC	-	-	-	-
#BC	$4\times 64$	$6\times 64^{2}$	$8\times 16^{3}$	$10\times 16^{4}$
#Test	$64^{2}$	$64^{3}$	$16^{4}$	$16^{5}$
Epochs	50,000	80,000	100,000	200,000

Table 6: Training Settings for High-Dimensional Helmholtz equations.

Method	$R$	2D	3D	4D	5D
Vanilla	–	13.344	43.467	56.801	180.465
CP-PINN	–	14.421	28.907	42.602	63.602
PIXEL	8	14.276	41.424	59.750	OOM
	16	14.610	38.370	63.222	OOM
	32	14.153	38.310	65.291	OOM
H-Spline	8	11.364	36.530	OOM	OOM

Table 8: Training Time (ms./iter.) for High-Dimensional Helmholtz equations.

Method	$R$	2D	3D	4D	5D
Vanilla	–	2.03E+00	1.80E+00	4.29E+00	7.36E+00
CP-PINN	–	7.78E-02	1.64E-01	1.33E+00	2.72E+01
PIXEL	8	8.94E-01	4.22E-01	3.99E+00	OOM
	16	9.08E-01	9.01E-01	1.16E+00	OOM
	32	1.02E+00	1.07E+00	1.01E+00	OOM
H-Spline	8	1.33E+00	2.56E+00	OOM	OOM

Table 9: Relative Errors for High-Dimensional Helmholtz equations.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Interpolation kernels and their first and second derivative functions: (a) cosine and (b)–(d) Hermite interpolation kernels.

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Physics-informed neural networks (PINNs) have emerged as a promising mesh-free paradigm for solving partial differential equations, yet adoption in science and engineering is limited by slow training and modest accuracy relative to modern numerical solvers. We introduce the Sequential Correction Algorithm for Learning Efficient PINN (Scale-PINN), a learning strategy that bridges modern physics-informed learning with numerical algorithms. Scale-PINN incorporates the iterative residual-correction principle, a cornerstone of numerical solvers, directly into the loss formulation, marking a paradigm shift in how PINN losses can be conceived and constructed. This integration enables Scale-PINN to achieve unprecedented convergence speed across PDE problems from different physics domain, including reducing training time on a challenging fluid-dynamics problem for state-of-the-art PINN from hours to sub-2 minutes while maintaining superior accuracy, and enabling application to representative problems in aerodynamics and urban science. By uniting the rigor of numerical methods with the flexibility of deep learning, Scale-PINN marks a significant leap toward the practical adoption of PINNs in science and engineering through scalable, physics-informed learning. Codes are available at https://github.com/chiuph/SCALE-PINN.

Abstract

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