cs.LGAug 25, 2026

Physics-Informed Error Field Learning: A Post-Training Optimization Framework for Physics-Informed Neural Networks

Authors: Jiuyun Sun, Yong Zhang

Organizations: College of Mathematics and Systems Science, Shandong University of Science and Technology, Qianwangang Road 579, Qingdao, 266590, Shandong, China

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Abstract

Physics-Informed Neural Networks (PINNs) have emerged as an important class of numerical methods for solving partial differential equations (PDEs). However, during the late-stage optimization process, further parameter updates often yield diminishing accuracy improvements while increasing computational costs. To address this issue, this paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for PINNs. Unlike conventional approaches that continuously approximate the solution field using a single network, PIEFL introduces an auxiliary error network after the primary network achieves satisfactory accuracy and shifts the learning objective from the solution field to the error field. By deriving error control equations under physical constraints, the error network learns the discrepancy between the current approximation and the exact solution, and the learned error correction is combined with the primary prediction to improve solution accuracy. The proposed framework avoids continuous optimization of the entire solution space and focuses computational resources on correcting existing prediction errors. Moreover, PIEFL requires no modification to the primary network architecture, making it compatible with existing PINN models and applicable as a general post-training optimization strategy. Numerical experiments on representative PDEs demonstrate that PIEFL achieves higher solution accuracy under the same computational budget, validating its effectiveness in improving the performance of PINNs.

Figures & tables

Figure 1: Schematic diagram of PIEFL
Figure 2: Solutions of the KdV equation: (a) Exact solution; (b) Prediction of the primary network; (c) Prediction of the standard PINN model; (d) Prediction of the PIEFL model.
Figure 3: Error fields of the KdV equation: (a) Error field ueu_{e} ; (b) Correction field u^e\hat{u}_{e} .
Figure 4: Evolution of the relative errors of PINNs, PIEFL, and the error network during the late-stage training.
α\alpha10.10.010.0010.0001
Error( u^e\hat{u}_{e} )7.483748e−17.483748\mathrm{e}{-1}6.965753e−16.965753\mathrm{e}{-1}4.498170e−24.498170\mathrm{e}{-2}3.737533e−23.737533\mathrm{e}{-2}3.805477e−23.805477\mathrm{e}{-2}
Error( ufinalu_{final} )2.878297e−32.878297\mathrm{e}{-3}2.679072e−32.679072\mathrm{e}{-3}1.730028e−41.730028\mathrm{e}{-4}1.437481e−41.437481\mathrm{e}{-4}1.463611e−41.463611\mathrm{e}{-4}
ERR−203.60%-203.60{\%}−182.58%-182.58{\%}81.75 %{\%}84.83 %{\%}84.56 %{\%}
Table 1: KdV equation: Relative L2\mathbb{L}_{2} errors between the predicted and exact solutions for different scaling coefficients
solutionprimary networkerror networkPIEFLPINNs(10000 iter)ERR
one-soliton solution2.773287e−32.773287\mathrm{e}{-3}2.105211e−32.105211\mathrm{e}{-3}8.745380e−58.745380\mathrm{e}{-5}1.829887e−31.829887\mathrm{e}{-3}95.22%95.22\%
two-soliton solution1.086328e−21.086328\mathrm{e}{-2}2.592568e−22.592568\mathrm{e}{-2}3.573394e−43.573394\mathrm{e}{-4}4.199116e−34.199116\mathrm{e}{-3}91.49%91.49\%
Table 2: Nonlinear Schrödinger equation: Relative L2\mathbb{L}_{2} errors of one-soliton and two-soliton solutions
Figure 5: The one-soliton solution of the nonlinear Schrödinger equation: (a) Exact solution; (b) Prediction of the primary network; (c) Prediction of the standard PINN model; (d) Prediction of the PIEFL model.
Figure 6: Error fields of the one-soliton solution of the nonlinear Schrödinger equation: (a) Error field qeq_{e} ; (b) Correction field q^e\hat{q}_{e} .
Figure 7: The two-soliton solution of the nonlinear Schrödinger equation: (a) Exact solution; (b) Prediction of the primary network; (c) Prediction of the standard PINN model; (d) Prediction of the PIEFL model.
Figure 8: Error fields of the two-soliton solution of the nonlinear Schrödinger equation: (a) Error field qeq_{e} ; (b) Correction field q^e\hat{q}_{e} .
solutionprimary networkerror networkPIEFLPINNsERR
one-soliton solution1.254043e−21.254043\mathrm{e}{-2}7.678907e−27.678907\mathrm{e}{-2}9.629841e−49.629841\mathrm{e}{-4}5.039722e−35.039722\mathrm{e}{-3}80.89%80.89\%
two-soliton solution1.493969e−21.493969\mathrm{e}{-2}3.923684e−13.923684\mathrm{e}{-1}5.861866e−35.861866\mathrm{e}{-3}9.5369942e−39.5369942\mathrm{e}{-3}38.53 %\%
Table 3: KP equation: Relative L2\mathbb{L}_{2} errors of one-soliton and two-soliton solutions
Figure 9: The one-soliton solution of the KP equation: (a) Exact solution; (b) Predicted solution of the primary network; (c) Predicted solution of the PINNs; (d) Predicted solution of the PIEFL
Figure 10: The one-soliton solution of the KP equation: (a) The error field ueu_{e} ; (b) The correction field u^e\hat{u}_{e}
Figure 11: The two-soliton solution of the KP equation: (a) Exact solution; (b) Predicted solution of the primary network; (c) Predicted solution of the PINNs; (d) Predicted solution of the PIEFL
Figure 12: The two-soliton solution of the KP equation: (a) The error field ueu_{e} ; (b) The correction field u^e\hat{u}_{e}
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