Mathematical theory of the attention mechanism and its applications in operator learning
Blog posts
References

Guo, R., Cao, S. and Chen, L. (2022) “Transformer Meets Boundary Value Inverse Problems.”

Cao, S. (2021) “Choose a Transformer: Fourier or Galerkin,” in 35th Conference on Neural Information Processing Systems (NeurIPS 2021). Available at: https://openreview.net/forum?id=ssohLcmn4r.
Theory and applications of virtual element method for interface problems
Blog posts
 Implementing data structures for polygonal finite elements
 Remarks and notes on the trace theorems on Lipschitz domains
References

Cao, S., Chen, L., Guo, R. and Lin, F. (2022) “Immersed Virtual Element Methods for Elliptic Interface Problems in Two Dimensions,” Journal of Scientific Computing, 93(1), pp. 1–41. doi: 10.1007/s1091502201949x.

Cao, S., Chen, L. and Guo, R. (2021) “A Virtual Finite Element Method for TwoDimensional Maxwell Interface Problems with a Background Unfitted Mesh,” Mathematical Models and Methods in Applied Sciences (M3AS), 31(14), pp. 2907–2936. doi: 10.1142/S0218202521500652.

Cao, S. (2021) “A simple virtual elementbased flux recovery on quadtree,” Electronic Research Archive, 29(6), pp. 3629–3647.

Cao, S. and Chen, L. (2019) “Anisotropic error estimates of the linear nonconforming virtual element methods,” SIAM Journal on Numerical Analysis. SIAM, 57(3), pp. 1058–1081.

Cao, S. and Chen, L. (2018) “Anisotropic error estimates of the linear virtual element method on polygonal meshes,” SIAM Journal on Numerical Analysis. SIAM, 56(5), pp. 2913–2939.
Acknowledgement
 This research was supported in part by the National Science Foundation under grants DMS1913080 and DMS2136075.
Finite element methods for problems related to Maxwell’s equations
Blog posts
 Notes on the explicit construction of a Helmholtz decomposition
 Boundary conditions for a twodimensional toy problem
 Deriving the expression of the Lorentz force using the language of a mathematician
References

Cao, S., Chen, L. and Huang, X. (2021) “Error analysis of a decoupled finite element method for quadcurl problems,” Journal of Scientific Computing, 90(1). doi: 10.1007/s10915021017057.

Cao, S., Wang, C. and Wang, J. (2022) “A New Numerical Method for DivCurl Systems with Low Regularity Assumptions,” Computers & Mathematics with Applications, 114, pp. 47–59. doi: 10.1016/j.camwa.2022.03.015.

Cai, Z., Cao, S. and Falgout, R. (2016) “Robust a posteriori error estimation for finite element approximation to H (curl) problem,” Computer Methods in Applied Mechanics and Engineering. Elsevier, 309, pp. 182–201.

Cai, Z. and Cao, S. (2015) “A recoverybased a posteriori error estimator for H(curl) interface problems,” Computer Methods in Applied Mechanics and Engineering. Elsevier, 296, pp. 169–195.
Finite element methods and other miscellaneous topics
Blog posts
 iFEM project: construction and a MATLAB implementation of the hierarchical basis for linear finite elements
 Kernels of a bounded operator’s extension
 The wellposedness of Robin boundary value problems
 An example on the limiting case of Sobolev embedding
 The little warming up project: the discontinuous Galerkin method for linear advection equation