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Abstract This work analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation, which are important partial differential equations (PDEs) in various applications. Different from previous works, which study the radius of convergence for the forward and inverse scattering series, the stability, and the approximation error of the series under theL^pnorms, we study these quantities under the SobolevH^snorm, which associates with a general class ofL²-based function spaces. TheH^snorm has a natural spectral bias based on its definition in the Fourier domain: the cases < 0 biases towards the lower frequencies, while the cases > 0 biases towards the higher frequencies. We compare the stability estimates using differentH^snorms for both the parameter and data domains and provide a theoretical justification for the frequency weighting techniques in practical inversion procedures. We also provide numerical inversion examples to demonstrate the differences in the inverse scattering radius of convergence under different metric spaces. 

Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative that requires minimal hyperparameter tuning and scales favorably to high-dimensional problems. In particular, we use a projection-based optimal transport solver [Meng et al.,Advances in Neural Information Processing Systems (Curran Associates, 2019), Vol. 32] to join successive samples and, subsequently, use transport splines (Chewi et al., 2020) to interpolate the evolving density. When the sampling frequency is sufficiently high, the optimal maps are close to the identity and are, thus, computationally efficient to compute. Moreover, the training process is highly parallelizable as all optimal maps are independent and can, thus, be learned simultaneously. Finally, the approach is based solely on numerical linear algebra rather than minimizing a nonconvex objective function, allowing us to easily analyze and control the algorithm. We present several numerical experiments on both synthetic and real-world datasets to demonstrate the efficiency of our method. In particular, these experiments show that the proposed approach is highly competitive compared with state-of-the-art normalizing flows conditioned on time across a wide range of dimensionalities. 

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269–310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.