cs.LGNov 29, 2022

FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation

Authors: Adarsh Ganeshram, Valentin Duruisseaux, Haydn Maust, Zongyi Li, Yixuan Wang, Daniel Leibovici, Oscar Bruno, Thomas Hou, 

Organizations: Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

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Abstract

The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution operators in function spaces and leverages physics losses during training to penalize deviations from known physics laws. Spectral differentiation provides an efficient way to compute derivatives for the physics losses, but it inherently assumes periodicity. When applied to non-periodic functions, this assumption can lead to significant errors, including Gibbs phenomena near domain boundaries which degrade the accuracy of both function representations and derivative computations. To overcome this limitation, we introduce the FC-PINO (Fourier-Continuation-based Physics-Informed Neural Operator) architecture which extends the accuracy and efficiency of PINO and spectral differentiation to non-periodic and non-smooth PDEs. In FC-PINO, we propose integrating Fourier continuation into the PINO framework, and test two different continuation approaches: FC-Legendre and FC-Gram. By transforming non-periodic signals into periodic functions on extended domains in a well-conditioned manner, Fourier continuation enables fast and accurate derivative computations. This approach avoids the discretization sensitivity of finite differences and the memory overhead of automatic differentiation. We demonstrate that standard PINO fails (without padding) or struggles (even with padding) to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks. In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives, and demonstrating that Fourier continuation is critical for extending PINO to a wider range of PDE problems when high-precision solutions are needed.

Figures & tables

Figure 1: Examples of periodic extensions obtained using the Fourier continuation methods. (top) Extensions of f(x)=sin⁡(16x)−cos⁡(8x)f(x)=\sin(16x)-\cos(8x) from [0,1][0,1] to [−0.25,1.25][-0.25,1.25] . (bottom) Extensions of f(x,y)=sin⁡(12x)−cos⁡(14y)+3xyf(x,y)=\sin(12x)-\cos(14y)+3xy from [0,1]2[0,1]^{2} to [−0.25,1.25]2[-0.25,1.25]^{2} .
Figure 2: The Fourier Neural Operator (FNO) architecture (extracted from ( Li et al., 2020 ) ).
Figure 3: Visualization of the Gibbs phenomenon when applying spectral differentiation to a non-periodic solution U(y)U(y) to the 1D Burgers’ equation ( 14 ). ( top ) Applying spectral differentiation on the extended domain after using FC–Gram . ( bottom two ) Applying spectral differentiation on the original domain without Fourier continuation. We observe large spurious oscillations, whose magnitude grows rapidly with the derivative order.
Figure 4
Figure 5
Figure 4: Illustration of the various variables used in the Fourier continuation. The given signal ff of nn points given on [a,b][a,b] is extended to a periodic signal with (n+c)(n+c) points, and the extension is split and appended on both sides (each with c/2c/2 points). FC–Legendre and FC–Gram construct periodic extension using only the information contained in the boundary vectors fℓf_{\ell} and frf_{r} of width dd , and not the remaining part fintf_{\text{int}} of the signal.
ModelPDE ResidualBC LossSmoothnessTotal Loss
FC–Gram    
FC–PINO1.9×10−111.9\times 10^{-11}9.2×10−159.2\times 10^{-15}5.5×10−95.5\times 10^{-9}6.7×10−106.7\times 10^{-10}
FC–Legendre    
FC–PINO1.8×10−121.8\times 10^{-12}1.9×10−141.9\times 10^{-14}2.8×10−82.8\times 10^{-8}2.8×10−92.8\times 10^{-9}
Baselines    
Pad–PINO1.4×10−81.4\times 10^{-8}1.1×10−101.1\times 10^{-10}4.9×10−54.9\times 10^{-5}4.9×10−64.9\times 10^{-6}
Table 1: PDE residual, boundary condition (BC) loss, smoothness loss, and total loss for FC–PINO with FC–Gram and FC–Legendre , and baseline PINO architectures when solving the self-similar Burgers’ equation ( 14 ) for λ=0.5\lambda=0.5 .
Model ( FC–Gram)PDE ResidualBC LossSmoothness
λ=1/2\lambda=1/28.8×10−108.8\times 10^{-10}3.7×10−123.7\times 10^{-12}5.9×10−85.9\times 10^{-8}
λ=1/12\lambda=1/122.3×10−102.3\times 10^{-10}6.7×10−136.7\times 10^{-13}6.1×10−86.1\times 10^{-8}
λ=1/22\lambda=1/221.5×10−101.5\times 10^{-10}8.3×10−118.3\times 10^{-11}1.3×10−71.3\times 10^{-7}
λ=1/32\lambda=1/328.3×10−108.3\times 10^{-10}1.8×10−101.8\times 10^{-10}1.6×10−71.6\times 10^{-7}
λ=1/52\lambda=1/522.2×10−92.2\times 10^{-9}8.6×10−108.6\times 10^{-10}3.4×10−73.4\times 10^{-7}
Fine-tuning Average8.6×10−108.6\times 10^{-10}2.3×10−102.3\times 10^{-10}1.5×10−71.5\times 10^{-7}
Table 2: PDE residual, boundary condition (BC) loss, and smoothness loss when solving the self-similar Burgers’ equation with FC–PINO over a family of values of λ\lambda . Here, we report the errors obtained when fine-tuning the pretrained model on the individual values λ∈{1/2,1/12,1/22,1/32,1/52}\lambda\in\{1/2,1/12,1/22,1/32,1/52\} , and the average over these 5 instances.
Figure 5: PDE residual loss across different λ\lambda ’s when fine-tuning the pretrained FC–PINO . The black curve represents the pretrain phase where we use the Adam optimizer for 3000 epochs, learning a solution operator over multiple λ\lambda values. We subsequently switch to the L–BFGS optimizer, as signified by the dashed line, and fine-tune the operator individually on 5 different PDE instances corresponding to different values of λ\lambda .
ModelPDE ResidualIC LossBC Loss
FC–Gram   
FC–PINO1.35×10−71.35\times 10^{-7}7.46×10−7\bm{7.46\times 10^{-7}}5.09×10−75.09\times 10^{-7}
FC–Legendre   
FC–PINO1.17×10−7\bm{1.17\times 10^{-7}}8.25 ×10−78.25\ \times 10^{-7}5.06×10−7\bm{5.06\times 10^{-7}}
Baselines   
Pad–PINO2.8×10−52.8\times 10^{-5}7.0×10−67.0\times 10^{-6}5.3×10−75.3\times 10^{-7}
Table 3: PDE residual, initial condition (IC) and boundary condition (BC) losses for FC–PINO with FC–Gram and FC–Legendre , and baseline PINO architectures for the 2D Burgers’ equation ( 17 ).
ModelPDE ResidualIC LossBC Loss
FC–Gram   
FC–PINO7.5×10−77.5\times 10^{-7}4.5×10−74.5\times 10^{-7}6.7×10−76.7\times 10^{-7}
FC–Legendre   
FC–PINO1.1×10−61.1\times 10^{-6}1.4×10−61.4\times 10^{-6}1.1×10−61.1\times 10^{-6}
Baselines   
Pad–PINO8.0×10−68.0\times 10^{-6}9.2×10−69.2\times 10^{-6}3.5×10−63.5\times 10^{-6}
Table 4: PDE residual, initial condition (IC) loss, and boundary condition (BC) loss for FC–PINO with FC–Gram and FC–Legendre , and baseline PINO architectures for the Navier–Stokes equations ( 19 ).
Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 12
Figure 13
ModelPDE ResidualBC LossSmoothnessTotal Loss
FC–Gram    
FC–PINO1.9×10−111.9\times 10^{-11}9.2×10−159.2\times 10^{-15}5.5×10−95.5\times 10^{-9}6.7×10−106.7\times 10^{-10}
PINO In–FC7.9×10−137.9\times 10^{-13}2.9×10−142.9\times 10^{-14}8.3×10−88.3\times 10^{-8}8.3×10−98.3\times 10^{-9}
PINO Out–FC2.1×10−122.1\times 10^{-12}2.1×10−132.1\times 10^{-13}1.3×10−71.3\times 10^{-7}1.3×10−81.3\times 10^{-8}
FC–Legendre    
FC–PINO1.8×10−12\bm{1.8\times 10^{-12}}2.2×10−15\bm{2.2\times 10^{-15}}2.8×10−8\bm{2.8\times 10^{-8}}2.8×10−9\bm{2.8\times 10^{-9}}
Appendix
Table 5: PDE residual, boundary condition (BC) loss, smoothness loss, and total loss for the FC-based PINO architectures using FC–Legendre and FC–Gram when solving the 1D Burgers’ equation for λ=0.5\lambda=0.5 .
ModelPDE ResidualIC LossBC Loss
FC–Gram   
FC–PINO1.35×10−7\bm{1.35\times 10^{-7}}7.46×10−7\bm{7.46\times 10^{-7}}5.09×10−7\bm{5.09\times 10^{-7}}
PINO In–FC1.73×10−71.73\times 10^{-7}7.55×10−77.55\times 10^{-7}5.21×10−75.21\times 10^{-7}
PINO Out–FC3.48×10−53.48\times 10^{-5}6.89×10−66.89\times 10^{-6}6.21×10−76.21\times 10^{-7}
FC–Legendre   
FC–PINO1.17×10−7\bm{1.17\times 10^{-7}}8.25×10−7\bm{8.25\times 10^{-7}}5.06×10−7\bm{5.06\times 10^{-7}}
Appendix
Table 6: PDE residual, initial condition (IC) loss, and boundary condition (BC) loss for the FC-based PINO architectures using FC–Legendre and FC–Gram when solving the 2D Burgers’ equation ( 17 ).
ModelPDE ResidualIC LossBC Loss
FC–Gram   
FC–PINO7.5×10−77.5\times 10^{-7}4.5×10−74.5\times 10^{-7}6.7×10−76.7\times 10^{-7}
PINO In–FC1.3×10−61.3\times 10^{-6}9.6×10−79.6\times 10^{-7}1.9×10−61.9\times 10^{-6}
PINO Out–FC1.3×10−51.3\times 10^{-5}1.7×10−61.7\times 10^{-6}7.2×10−67.2\times 10^{-6}
FC–Legendre   
FC–PINO1.1×10−6\bm{1.1\times 10^{-6}}1.4×10−61.4\times 10^{-6}1.1×10−6\bm{1.1\times 10^{-6}}
Appendix
Table 7: PDE residual, initial condition (IC) loss, and boundary condition (BC) loss for the FC-based PINO architectures using FC–Legendre and FC–Gram when solving the Navier–Stokes equations ( 19 ).
Figure 6: Effect of varying continuation parameters cc (continuation points) and d (degree) for FC–Legendre on Total Loss (left) and PDE Residual (right)
 Total LossPDE Residual
d=2d=21.05×10−81.05\times 10^{-8}3.74×10−103.74\times 10^{-10}
d=3d=34.89×10−94.89\times 10^{-9}9.52×10−109.52\times 10^{-10}
d=4d=47.34×10−107.34\times 10^{-10}1.25×10−101.25\times 10^{-10}
d=5d=58.29×10−108.29\times 10^{-10}1.12×10−91.12\times 10^{-9}
d=6d=66.72×10−10\mathbf{6.72\times 10^{-10}}1.92×10−11\mathbf{1.92\times 10^{-11}}
d=7d=76.60×10−96.60\times 10^{-9}1.07×10−91.07\times 10^{-9}
Appendix
Table 8: Effect of the choice of continuation parameters dd for FC–Gram on the Total Loss and PDE Residual when using FC–PINO to solve the 1D Burgers’ equation ( 14 ).
ModelFC MethodddccFourier ModesLearning Rate
FC–PINOFC–Gram650240.0001
FC–Legendre470480.001
PINO In–FCFC–Gram450240.005
FC–Legendre570240.0001
PINO Out–FCFC–Gram450240.001
FC–Legendre650480.005
Appendix
Table 9: Optimal Hyperparameters for the PINO models for the 1D Burgers equation.
ModelFC MethodddccLearning Rate
FC–PINOFC–Gram4500.0001
FC–Legendre6300.001
PINO In–FCFC–Gram6500.0001
FC–Legendre6400.001
PINO Out–FCFC–Gram6500.0001
FC–Legendre6800.001
Appendix
Table 10: Optimal Hyperparameters for the PINO models for the 2D Burgers equation.
ModelFC MethodddccSpatial ModesTemporal Modes
FC–PINOFC–Gram8503220
FC–Legendre7403220
PINO In–FCFC–Gram7503224
FC–Legendre8603224
PINO Out–FCFC–Gram8503220
FC–Legendre8603224
Appendix
Table 11: Optimal Hyperparameters for the PINO models for the 3D Navier–Stokes equation.
Figure 7: Predicted solutions of PINO architectures for the 1D Burger’s equation.
Figure 8: Predicted solutions when fine-tuning pretrained FC–PINO models on different λ\lambda values for the 1D Burgers’ equation.
Figure 9: Predicted solutions ( top five ) and corresponding PDE residuals ( bottom five ) for baseline PINO architectures and FC–PINO variants for the 2D Burgers’ equation.
Figure 10: Predicted solutions ( top five ) and corresponding PDE residuals ( bottom five ) for baseline PINO architectures and FC–PINO variants for the horizontal velocity component of the Navier–Stokes equations.
Figure 11: Predicted solutions ( top five ) and corresponding PDE residuals ( bottom five ) for baseline PINO architectures and FC–PINO variants for the horizontal velocity component of the Navier–Stokes equations.
Figure 12: Evolution of the PDE residual loss for the 1D Burgers’ equation with FC–PINO .
Figure 13: Evolution of the PDE residual loss for the 2D Burgers’ equation with FC–PINO .
Figure 14: Evolution of the PDE residual for the Navier–Stokes equations with FC–PINO .
Figure 15: Evolution of the total loss as a function of time for the 1D Burgers’ equation with the different PINO approaches. To allow for a cleaner visualization, we display the time taken to reach certain threshold loss values for the first time as a solid line. The dotted line then connects the loss at the last threshold value and the loss at 360360 seconds.
Figure 16: Evolution of the PDE residual loss as a function of time for the 2D Burgers’ equation with the different PINO approaches. To allow for a cleaner visualization, we display the time taken to reach certain threshold loss values for the first time as a solid line. The dotted line then connects the loss at the last threshold value and the loss at 6060 minutes.
Figure 17: Evolution of the PDE residual as a function of time for the Navier–Stokes equations with the different PINO approaches. To allow for a cleaner visualization, we display the time taken to reach certain threshold loss values for the first time as a solid line. The dotted line then connects the loss at the last threshold value and the final loss at the final value.
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