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

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References

  1. Guo, R., Cao, S. and Chen, L. (2022) “Transformer Meets Boundary Value Inverse Problems.”
  2. 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=ssohLcmn4-r.

   

Theory and applications of virtual element method for interface problems

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References

  1. 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/s10915-022-01949-x.
  2. Cao, S., Chen, L. and Guo, R. (2021) “A Virtual Finite Element Method for Two-Dimensional 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.
  3. Cao, S. (2021) “A simple virtual element-based flux recovery on quadtree,” Electronic Research Archive, 29(6), pp. 3629–3647.
  4. 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.
  5. 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 DMS-1913080 and DMS-2136075.

   

Blog posts

References

  1. Cao, S., Chen, L. and Huang, X. (2021) “Error analysis of a decoupled finite element method for quad-curl problems,” Journal of Scientific Computing, 90(1). doi: 10.1007/s10915-021-01705-7.
  2. Cao, S., Wang, C. and Wang, J. (2022) “A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions,” Computers & Mathematics with Applications, 114, pp. 47–59. doi: 10.1016/j.camwa.2022.03.015.
  3. 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.
  4. Cai, Z. and Cao, S. (2015) “A recovery-based 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

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