5:15 PM - 6:45 PM
[HDS11-P08] Data Assimilation of Tsunami using Physics-Informed Neural Network (PINN)
Keywords:tsunami, PINN, data assmilation
1. Background and Objectives
With the development of dense tsunami observation networks such as S-net, it has become practical to directly estimate the wavefield from observed data without estimating the initial wave source. Data assimilation techniques for tsunamis proposed so far, such as Optimal Interpolation method (Maeda et al. 2015 GRL) and Adjoint method (Maeda 2023 SSJ), are based on the FDM calculation of tsunami equations and estimate the optimal values of physical quantities defined at grid points over time.
In this study, we propose a new method to evaluate tsunami wavefields using Neural Networks (NN). NNs have the ability to approximate continuous functions and can reconstruct actual wavefields by interpolating observed data at limited points. However, simple interpolation cannot accurately assimilate in regions with sparse data. Therefore, in this study, we conduct training of NNs considering the physical laws of tsunami propagation using Physics-Informed Neural Network (PINN, Raissi et al. 2019 JCP). This is expected to improve assimilation accuracy compared to simple NNs trained with only data.
2. Method
In this experiment, we trained PINN using synthetic waveforms obtained by tsunami propagation simulations (JAGURS, Baba et al. 2015 PAGEOPH). We also used the synthetic wavefield by JAGURS as the "ground truth" wavefield that is necessary to evaluate the estimated wavefield by PINN.
The PINN used for assimilation and prediction receives spatiotemporal coordinates (x, y, t) as inputs, and outputs wave height ζ and integrated fluxes (M, N) (Fig. 1). For the loss function, we considered
(a) L_data: the sum of squared errors between the observed wave heights and the estimated values by PINN at all observation points (constraining PINN in terms of data)
(b) L_PDE: the sum of squared residuals of the tsunami equations calculated at Collocation Points in the (x, y, t) space (physically constraining PINN)
(c) L_BC: constraint to satisfy the boundary conditions of M=N=0 at coastal points
We trained PINN to minimize the sum of (a), (b), and (c). The bathymetry D(x, y) in the tsunami equation was given as known, and set to D=0 on land.
Using the trained NN, we calculated the estimated waveforms at each observation point and wavefields at each time, which were used for the performance evaluation of PINN.
Python library DeepXDE (Lu et al. 2021 SIAM review) was used for the design of PINN.
3. Results and Discussion
Based on the tsunami wave source model of the 2011 Tohoku-Oki earthquake, we conducted experiments on assimilation and prediction of tsunami wavefields using simulated waveforms at S-net and tide gauge stations. For PINN training, we used wave height data at observation points and tsunami flux data (M=N=0) at dummy observation points along the coastline. To stabilize the calculations, dummy observation points were placed on land which always obtain data with wave height 0.
(1) Assimilation Experiment
First, we attempted assimilation of wavefields using observation data during t = 0 - 60 min. (sampling interval : 30 sec.). As a result, simple NN (only data interpolation) could reproduce tsunami wave heights at observation points, but non-physical waves appeared outside the observation points (Fig. 2 middle). In contrast, PINN could reproduce smooth tsunami wavefields consistent with the ground truth (Fig. 2 bottom). However, even with physical constraints, assimilation of complex reflected and scattered waves near the coast is challenging.
(2) Prediction Experiment
Next, using only initial (t=0-10 min.) observation data (sampling interval : 10 sec.), we attempted prediction of wavefields for subsequent times (up to t=40 min.). The propagation process of offshore tsunamis could be accurately predicted, but discrepancies were observed near the coast compared to the actual tsunami (Fig. 3 bottom). Properly addressing the complex coastal topography as PINN constraints is a future challenge.
4. Future Prospects
The time required for PINN training in this experiment was approximately 20-30 minutes on the Wisteria-a supercomputer (8 GPU parallel), which is comparable to the computational cost of conventional data assimilation methods (calculation of estimated waveforms using trained PINN can be instantly performed on CPU). Furthermore, by starting the optimization of NN weights from NN weights representing another tsunami event (transfer learning) instead of starting from the initialized weights, further acceleration may be achievable.
Since the numerical experiments in this study used simulated waveforms, we plan to perform experiments with observed data (e.g., 2016 Off-Fukushima tsunami) in the future.
With the development of dense tsunami observation networks such as S-net, it has become practical to directly estimate the wavefield from observed data without estimating the initial wave source. Data assimilation techniques for tsunamis proposed so far, such as Optimal Interpolation method (Maeda et al. 2015 GRL) and Adjoint method (Maeda 2023 SSJ), are based on the FDM calculation of tsunami equations and estimate the optimal values of physical quantities defined at grid points over time.
In this study, we propose a new method to evaluate tsunami wavefields using Neural Networks (NN). NNs have the ability to approximate continuous functions and can reconstruct actual wavefields by interpolating observed data at limited points. However, simple interpolation cannot accurately assimilate in regions with sparse data. Therefore, in this study, we conduct training of NNs considering the physical laws of tsunami propagation using Physics-Informed Neural Network (PINN, Raissi et al. 2019 JCP). This is expected to improve assimilation accuracy compared to simple NNs trained with only data.
2. Method
In this experiment, we trained PINN using synthetic waveforms obtained by tsunami propagation simulations (JAGURS, Baba et al. 2015 PAGEOPH). We also used the synthetic wavefield by JAGURS as the "ground truth" wavefield that is necessary to evaluate the estimated wavefield by PINN.
The PINN used for assimilation and prediction receives spatiotemporal coordinates (x, y, t) as inputs, and outputs wave height ζ and integrated fluxes (M, N) (Fig. 1). For the loss function, we considered
(a) L_data: the sum of squared errors between the observed wave heights and the estimated values by PINN at all observation points (constraining PINN in terms of data)
(b) L_PDE: the sum of squared residuals of the tsunami equations calculated at Collocation Points in the (x, y, t) space (physically constraining PINN)
(c) L_BC: constraint to satisfy the boundary conditions of M=N=0 at coastal points
We trained PINN to minimize the sum of (a), (b), and (c). The bathymetry D(x, y) in the tsunami equation was given as known, and set to D=0 on land.
Using the trained NN, we calculated the estimated waveforms at each observation point and wavefields at each time, which were used for the performance evaluation of PINN.
Python library DeepXDE (Lu et al. 2021 SIAM review) was used for the design of PINN.
3. Results and Discussion
Based on the tsunami wave source model of the 2011 Tohoku-Oki earthquake, we conducted experiments on assimilation and prediction of tsunami wavefields using simulated waveforms at S-net and tide gauge stations. For PINN training, we used wave height data at observation points and tsunami flux data (M=N=0) at dummy observation points along the coastline. To stabilize the calculations, dummy observation points were placed on land which always obtain data with wave height 0.
(1) Assimilation Experiment
First, we attempted assimilation of wavefields using observation data during t = 0 - 60 min. (sampling interval : 30 sec.). As a result, simple NN (only data interpolation) could reproduce tsunami wave heights at observation points, but non-physical waves appeared outside the observation points (Fig. 2 middle). In contrast, PINN could reproduce smooth tsunami wavefields consistent with the ground truth (Fig. 2 bottom). However, even with physical constraints, assimilation of complex reflected and scattered waves near the coast is challenging.
(2) Prediction Experiment
Next, using only initial (t=0-10 min.) observation data (sampling interval : 10 sec.), we attempted prediction of wavefields for subsequent times (up to t=40 min.). The propagation process of offshore tsunamis could be accurately predicted, but discrepancies were observed near the coast compared to the actual tsunami (Fig. 3 bottom). Properly addressing the complex coastal topography as PINN constraints is a future challenge.
4. Future Prospects
The time required for PINN training in this experiment was approximately 20-30 minutes on the Wisteria-a supercomputer (8 GPU parallel), which is comparable to the computational cost of conventional data assimilation methods (calculation of estimated waveforms using trained PINN can be instantly performed on CPU). Furthermore, by starting the optimization of NN weights from NN weights representing another tsunami event (transfer learning) instead of starting from the initialized weights, further acceleration may be achievable.
Since the numerical experiments in this study used simulated waveforms, we plan to perform experiments with observed data (e.g., 2016 Off-Fukushima tsunami) in the future.