线性和非线性扩散问题的深谱计算
Deep spectral computations in linear and nonlinear diffusion problems
摘要：
我们提出了一个灵活的机器学习框架解决扩散算子的本征值问题在适度大的维度。我们改进了现有的神经网络（NN）特征值求解器，证明我们的方法计算能力（i）非自伴算子的特征值解与小扩散（ii）本征对位于频谱深处（iii）计算几个本征模一次（iv）处理非线性本征值问题。要做到这一点，我们采用了变分方法，包括最小化的自然成本功能，涉及瑞利折射率，通过简单的绝热技术和多值前馈神经参数化的解决方案。令人信服的成功报道了一个10维本征值问题对应的Kolmogorov算子与混合Stepanov流。此外，我们表明，该方法允许提供准确的本征解的5-D薛定谔算子具有$32$亚稳态。此外，我们解决了所谓的Gelfand超线性问题具有指数非线性，在尺寸$4$，并为非平凡域表现出腔。特别是，我们得到NN-近似的高能量的解决方案接近奇异的。我们强调，这些结果中的每一个都是在经典方法由于维数灾难而无望的情况下使用小尺寸神经网络获得的。这一工作为研究Ruelle-Pollicot共振、降维、非线性本征值问题以及动力学无势时的亚稳性等问题提供了新的视角。
英文摘要：
We propose a flexible machine-learning framework for solving eigenvalue problems of diffusion operators in moderately large dimension. We improve on existing Neural Networks (NNs) eigensolvers by demonstrating our approach ability to compute (i) eigensolutions for non-self adjoint operators with small diffusion (ii) eigenpairs located deep within the spectrum (iii) computing several eigenmodes at once (iv) handling nonlinear eigenvalue problems. To do so, we adopt a variational approach consisting of minimizing a natural cost functional involving Rayleigh quotients, by means of simple adiabatic technics and multivalued feedforward neural parametrisation of the solutions. Compelling successes are reported for a 10-dimensional eigenvalue problem corresponding to a Kolmogorov operator associated with a mixing Stepanov flow. We moreover show that the approach allows for providing accurate eigensolutions for a 5-D Schrödinger operator having $32$ metastable states. In addition, we address the so-called Gelfand superlinear problem having exponential nonlinearities, in dimension $4$, and for nontrivial domains exhibiting cavities. In particular, we obtain NN-approximations of high-energy solutions approaching singular ones. We stress that each of these results are obtained using small-size neural networks in situations where classical methods are hopeless due to the curse of dimensionality. This work brings new perspectives for the study of Ruelle-Pollicot resonances, dimension reduction, nonlinear eigenvalue problems, and the study of metastability when the dynamics has no potential.
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