We extend the WSINDy (Weak SINDy) method of sparse recovery introduced previously by the authors ( arXiv:2005.04339) to the setting of partial differential equations (PDEs). As in the case of ODE discovery, the weak form replaces pointwise approximation of derivatives with local integrations against test functions and achieves effective machine-precision recovery of weights from noise-free data (i.e. below the tolerance of the simulation scheme) as well as natural robustness to noise without the use of noise filtering . The resulting WSINDy _PDE algorithm uses separable test functions implemented efficiently via convolutions for discovery of PDE models with computational complexity O(NM)from data points with M = N ^{D+1(N^{D+1}\log(N)) for datasets with N points } in each of D+1 dimensions . We demonstrate on several notoriously challenging PDEsthe speed and accuracy with which WSINDy_PDE recovers the correct models from datasets with surprisingly large levels noise (often with levels of noise much greater than 10\%)(\log(N)) operations per datapoint). Furthermore, our Fourier-based implementation reveals a connection between robustness to noise and the spectra of test functions, which we utilize in an }  .
Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data (Brunton et al., PNAS, '16; Rudy et al., Sci. Adv. '17). Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in ( arXiv:2005.04339) to the setting of partial differential equations (PDEs). The elimination of pointwise derivative approximations via the weak form enables effective machine-precision recovery of model coefficients from noise-free data (i.e. below the tolerance of the simulation scheme) as well as robust identification of PDEs in the large noise regime (with signal-to-noise ratio approaching one in many well-known cases). This is accomplished by discretizing a convolutional weak form of the PDE and exploiting separability of test functions for efficient model identification using the Fast Fourier Transform . The resulting WSINDy algorithm for PDEs has a worst-case computational complexity of \mathcal{O(N^{D+1}\log(N)) for datasets with N points } in each of D+1 dimensions (i.e. \mathcal{O(\log(N)) operations per datapoint). Furthermore, our Fourier-based implementation reveals a connection between robustness to noise and the spectra of test functions, which we utilize in an }a priori selection algorithm for test functions. Finally, we introduce a learning algorithm for the threshold in sequential-thresholding least-squares (STLS) that enables model identification from large libraries, and we utilize scale-invariance at the continuum level to identify PDEs from poorly-scaled datasets. We demonstrate WSINDy's robustness, speed and accuracy on several challenging PDEs .
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We extend the WSINDy (Weak SINDy) method of sparse recovery introduced previously by the authors (
Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data (Brunton et al., PNAS, '16; Rudy et al., Sci. Adv. '17). Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in (
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As in the case of ODE discovery,
The elimination of pointwise derivative approximations via
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replaces pointwise approximation of derivatives with local integrations against test functions and achieves
enables
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weights
model coefficients
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natural robustness to noise without the use of noise filtering
robust identification of PDEs in the large noise regime (with signal-to-noise ratio approaching one in many well-known cases). This is accomplished by discretizing a convolutional weak form of the PDE and exploiting separability of test functions for efficient model identification using the Fast Fourier Transform
R
_PDE algorithm uses separable test functions implemented efficiently via convolutions for discovery of PDE models with computational complexity O(NM)from data points with M = N ^{D+1
algorithm for PDEs has a worst-case computational complexity of \mathcal{O
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. We demonstrate on several notoriously challenging PDEsthe speed and accuracy with which WSINDy_PDE recovers the correct models from datasets with surprisingly large levels noise (often with levels of noise much greater than 10\%)
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a priori
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selection algorithm for test functions. Finally, we introduce a learning algorithm for the threshold in sequential-thresholding least-squares (STLS) that enables model identification from large libraries, and we utilize scale-invariance at the continuum level to identify PDEs from poorly-scaled datasets. We demonstrate WSINDy's robustness, speed and accuracy on several challenging PDEs