Publications

You can find an up-to-date list of my papers on my Google Scholar Page.

Papers Submitted/Under Review:

1. Lyu, Liyao, Xinyue Yu, and Hayden Schaeffer. "Multiscale Nudging: From Macroscopic Observations to Microscopic Dynamics."  arXiv preprint arXiv:2606.06809 (2026).

[Preprint]  

2. Weihs, Adrien and Hayden Schaeffer. "Multiple Neural Operator Achieve Near-Optimal Rates for Multi-Task Learning."  arXiv preprint arXiv:2605.22724 (2026).

[Preprint]  

3. Weihs, Adrien and Hayden Schaeffer. "Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks."  arXiv preprint arXiv:2604.01961 (2026).

[Preprint]  

4. Lyu, Liyao, Xinyue Yu, and Hayden Schaeffer. "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data."  arXiv preprint arXiv:2604.00333 (2026).

[Preprint]  

5. Zhang, Minxin, Yuxuan Liu, and Hayden Schaeffer. "Adam Improves Muon: Adaptive Moment Estimation with Orthogonalized Momentum."  arXiv preprint arXiv:2602.17080 (2026).

[Preprint]  

6. Zhu, Min, Jingmin Sun, Zecheng Zhang, Hayden Schaeffer, and Lu Lu. "PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations."  arXiv preprint arXiv:2512.23056 (2025).

[Preprint]  

7. Yu, Xinyue and  Hayden Schaeffer. "Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space."  arXiv preprint arXiv:2512.17884 (2025).

[Preprint]  

8. Zhang, Zecheng, Hao Liu, Guosheng Fu, Hayden Schaeffer, and Guang Lin. "Finite Element Representation Network (FERN) for Operator Learning with a Localized Trainable Basis"  arXiv preprint arXiv:2510.26962 (2025).

[Preprint]  

9. Weihs, Adrien, Jingmin Sun, Zecheng Zhang, and Hayden Schaeffer. "A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory."  arXiv preprint arXiv:2510.25379 (2025).

[Preprint]  

10. Zhang, Zecheng, Hao Liu, Guosheng Fu, Hayden Schaeffer, and Guang Lin. "Finite Element Representation Network (FERN) for operator Learning with a Localized Trainable Basis."  arXiv preprint arXiv:2510.26962 (2025).

[Preprint]  

11. Zhang, Minxin, Yuxuan Liu, and Hayden Schaeffer. "AdaGrad Meets Muon: Adaptive Stepsizes for Orthogonal Updates." arXiv preprint arXiv:2509.02981 (2025).

[Preprint]

12. Liao, Chunyang, Deanna Needell, and Hayden Schaeffer. "Cauchy Random Features for Operator Learning in Sobolev Space." arXiv preprint arXiv:2503.00300 (2025).

[Preprint]   [Code]

13. Negrini, Elisa, Yuxuan Liu, Liu Yang, Stanley J. Osher, and Hayden Schaeffer. "A Multimodal PDE Foundation Model for Prediction and Scientific Text Descriptions." arXiv preprint arXiv:2502.06026 (2025).

[Preprint]   [Code]

14. Liu, Yuxuan, Jingmin Sun, and Hayden Schaeffer. "BCAT: A Block Causal Transformer for PDE Foundation Models for Fluid Dynamics." arXiv preprint arXiv:2501.18972 (2025).

[Preprint]   [Code]

15. Zhang, Minxin, Fuqun Han, Yat Tin Chow, Stanley Osher, and Hayden Schaeffer. "Inexact Proximal Point Algorithms for Zeroth-Order Global Optimization." arXiv preprint arXiv:2412.11485 (2024).

[Preprint]   [Code]

16. Liao, Chunyang, Deanna Needell, Hayden Schaeffer, and Alexander Xue. "Differentially private random feature model." arXiv preprint arXiv:2412.04785 (2024).

[Preprint]   

17. Cao, Yadi, Yuxuan Liu, Liu Yang, Rose Yu, Hayden Schaeffer, and Stanley Osher. "VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction." arXiv preprint arXiv:2411.16063 (2024).

[Preprint]   [Code]

18. Zhang, Zecheng, Christian Moya, Lu Lu, Guang Lin, and Hayden Schaeffer. "DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuning." arXiv preprint arXiv:2411.07239 (2024).

[Preprint]

19. Liu, Hao, Zecheng Zhang, Wenjing Liao, and Hayden Schaeffer. "Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study." arXiv preprint arXiv:2410.00357 (2024).

[Preprint]

20. Sun, Jingmin, Zecheng Zhang, and Hayden Schaeffer. "LeMON: Learning to learn multi-operator networks." arXiv preprint arXiv:2408.16168 (2024).

[Preprint]   [Code]

Publications:

1. Zhang, Zecheng, Wing Tat Leung, and Hayden Schaeffer. "A discretization-invariant extension and analysis of some deep operator networks." Journal of Computational and Applied Mathematics 456 (2025): 116226.

[Preprint]   [Paper]   [Code]

2. Jollie, Derek, Jingmin Sun, Zecheng Zhang, and Hayden Schaeffer. "Time-series Forescasting and Refinement within a Multimodal PDE Foundation Model." Journal of Machine Learning for Modeling and Computing 6, no. 2 (2025): 77-89.

[Preprint]   [Paper]   [Code]

3. Liu, Yuxuan, Zecheng Zhang, and Hayden Schaeffer. "PROSE: Predicting multiple operators and symbolic expressions using multimodal transformers." Neural Networks 180 (2024): 106707.

[Preprint]   [Paper]   [Code]

4. Zhang, Zecheng, Christian Moya, Wing Tat Leung, Guang Lin, and Hayden Schaeffer. "Bayesian Deep Operator Learning for Homogenized to Fine-Scale Maps for Multiscale PDE." Multiscale Modeling & Simulation 22, no. 3 (2024): 956-972.

[Preprint]   [Paper]

5. Liu, Yuxuan, Jingmin Sun, Xinjie He, Griffin Pinney, Zecheng Zhang, and Hayden Schaeffer. "PROSE-FD: A multimodal pde foundation model for learning multiple operators for forecasting fluid dynamics." Neurips 2024 Workshop: Models for Science: Progress, Opportunities, and Challenges, (2024).

[Preprint]   [Paper]   [Code]   [Poster]

6. Zhang, Zecheng, Christian Moya, Lu Lu, Guang Lin, and Hayden Schaeffer. "D2NO: Efficient handling of heterogeneous input function spaces with distributed deep neural operators." Computer Methods in Applied Mechanics and Engineering 428 (2024): 117084.

[Preprint]   [Paper]

7. Chen, Zhijun, and Hayden Schaeffer. "Conditioning of random Fourier feature matrices: double descent and generalization error." Information and Inference: A Journal of the IMA 13, no. 2 (2024): iaad054.

[Preprint]   [Paper]

8. Richardson, Nicholas, Hayden Schaeffer, and Giang Tran. "SRMD: Sparse random mode decomposition." Communications on Applied Mathematics and Computation 6, no. 2 (2024): 879-906.

[Preprint]   [Paper]   [Code]

9. Sun, Jingmin, Yuxuan Liu, Zecheng Zhang, and Hayden Schaeffer. "Towards a foundation model for partial differential equations: Multioperator learning and extrapolation." Physical Review E 111, no. 3 (2025): 035304.

[Preprint]   [Paper]   [Code]

10. Saha, Esha, Hayden Schaeffer, and Giang Tran. "HARFE: hard-ridge random feature expansion." Sampling Theory, Signal Processing, and Data Analysis 21, no. 2 (2023): 27.

[Preprint]   [Paper]   [Code]

11. Zhang, Zecheng, Leung Wing Tat, and Hayden Schaeffer. "BelNet: Basis enhanced learning, a mesh-free neural operator." Proceedings of the Royal Society A 479, no. 2276 (2023): 20230043.

[Preprint]   [Paper]   [Code]

12. Liu, Yuxuan, Scott G. McCalla, and Hayden Schaeffer. "Random feature models for learning interacting dynamical systems." Proceedings of the Royal Society A 479, no. 2275 (2023): 20220835.

[Preprint]   [Paper]   [Code]

13. Hashemi, Abolfazl, Hayden Schaeffer, Robert Shi, Ufuk Topcu, Giang Tran, and Rachel Ward. "Generalization bounds for sparse random feature expansions." Applied and Computational Harmonic Analysis 62 (2023): 310-330.

[Preprint]   [Paper]   [Code]

14. Xie, Yuege, Robert Shi, Hayden Schaeffer, and Rachel Ward. "Shrimp: Sparser random feature models via iterative magnitude pruning." In Mathematical and Scientific Machine Learning, pp. 303-318. PMLR, 2022.

[Preprint]   [Paper]   [Code]

15. Chen, Zhijun, Hayden Schaeffer, and Rachel Ward. "Concentration of random feature matrices in high-dimensions." In Mathematical and Scientific Machine Learning, pp. 287-302. PMLR, 2022.

[Preprint]   [Paper]

16. Bollinger, Kayla, and Hayden Schaeffer. "Reduced order modeling using shallow ReLU networks with Grassmann layers." In Mathematical and Scientific Machine Learning, pp. 847-867. PMLR, 2022.

[Preprint]   [Paper]

17. Ho, Lam Si Tung, Hayden Schaeffer, Giang Tran, and Rachel Ward. "Recovery guarantees for polynomial coefficients from weakly dependent data with outliers." Journal of Approximation Theory 259 (2020): 105472.

[Preprint]   [Paper]

18. Sun, Yifan, Linan Zhang, and Hayden Schaeffer. "NeuPDE: Neural network based ordinary and partial differential equations for modeling time-dependent data." In Mathematical and Scientific Machine Learning, pp. 352-372. PMLR, 2020.

[Preprint]   [Paper]

19. Zhang, Linan, and Hayden Schaeffer. "Forward stability of ResNet and its variants." Journal of Mathematical Imaging and Vision 62 (2020): 328-351.

[Preprint]   [Paper]

20. Schaeffer, Hayden, Giang Tran, Rachel Ward, and Linan Zhang. "Extracting structured dynamical systems using sparse optimization with very few samples." Multiscale Modeling & Simulation 18, no. 4 (2020): 1435-1461.

[Preprint]   [Paper]   [Code]

21. Schaeffer, Hayden, and Scott G. McCalla. "Extending the step-size restriction for gradient descent to avoid strict saddle points." SIAM Journal on Mathematics of Data Science 2, no. 4 (2020): 1181-1197.

[Preprint]   [Paper]  

22. Zhang, Linan, and Hayden Schaeffer. "On the convergence of the SINDy algorithm." Multiscale Modeling & Simulation 17, no. 3 (2019): 948-972.

[Preprint]   [Paper]   [Code]

23. Zhang, Linan, and Hayden Schaeffer. "Stability and error estimates of BV solutions to the Abel inverse problem." Inverse Problems 34, no. 10 (2018): 105003.

[Preprint]   [Paper]

24. Schaeffer, Hayden, Giang Tran, and Rachel Ward. "Extracting sparse high-dimensional dynamics from limited data." SIAM Journal on Applied Mathematics 78, no. 6 (2018): 3279-3295.

[Preprint]   [Paper]   [Code]

25. Schaeffer, Hayden. "A penalty method for some nonlinear variational obstacle problems." Communications in Mathematical Sciences 16, no. 7 (2018): 1757-1777.

[Preprint]   [Paper]

26. Schaeffer, Hayden, and Scott G. McCalla. "Sparse model selection via integral terms." Physical Review E 96, no. 2 (2017): 023302.

[Paper]

27. Schaeffer, Hayden. "Learning partial differential equations via data discovery and sparse optimization." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2197 (2017): 20160446.

 [Paper]

28. Schaeffer, Hayden, and Thomas Y. Hou. "An accelerated method for nonlinear elliptic PDE." Journal of Scientific Computing 69 (2016): 556-580.

[Preprint]   [Paper]  

29. Hou, Thomas Y., Qin Li, and Hayden Schaeffer. "Sparse+ low-energy decomposition for viscous conservation laws." Journal of Computational Physics 288 (2015): 150-166.

[Paper]   

30. Tran, Giang, Hayden Schaeffer, William M. Feldman, and Stanley J. Osher. "An L^1 penalty method for general obstacle problems." SIAM Journal on Applied Mathematics 75, no. 4 (2015): 1424-1444.

[Preprint]   [Paper]

31. Schaeffer, Hayden, Yi Yang, and Stanley Osher. "Space-time regularization for video decompression." SIAM Journal on Imaging Sciences 8, no. 1 (2015): 373-402.

[Preprint]   [Paper]

32. Schaeffer, Hayden, Yi Yang, Hongkai Zhao, and Stanley Osher. "Real-time adaptive video compression." SIAM Journal on Scientific Computing 37, no. 6 (2015): B980-B1001.

[Preprint]   [Paper]

33. Caflisch, Russel E., Stanley J. Osher, Hayden Schaeffer, and Giang Tran. "PDEs with compressed solutions." Communications in Mathematical Sciences 13, no. 8 (2015): 2155-2176.

[Preprint]   [Paper] 

34. Ascenzi, Maria-Grazia, Xia Du, James I. Harding, Emily N. Beylerian, Brian M. de Silva, Ben J. Gross, Hannah K. Kastein, Weiguang Wang, Karen M. Lyons, and Hayden Schaeffer. "Automated Cell Detection and Morphometry on Growth Plate Images of Mouse Bone." Applied mathematics 5, no. 18 (2014): 2866.

[Paper]

35. Schaeffer, Hayden, and Luminita Vese. "Active contours with free endpoints." Journal of mathematical imaging and vision 49 (2014): 20-36.

[Preprint]   [Paper]

36. Schaeffer, Hayden, and Luminita Vese. "Variational dynamics of free triple junctions." Journal of Scientific Computing 59 (2014): 386-411.

[Paper]

37. Schaeffer, Hayden. "Active arcs and contours." Inverse Problems and Imaging 8, no. 3 (2014): 845-863.

[Preprint]   [Paper]

38. Schaeffer, Hayden, Nóirín Duggan, Carole le Guyader, and Luminita Vese. "Topology preserving active contours." Communications in Mathematical Sciences 12, no. 7 (2014): 1329-1342.

[Paper]

39. Mackey, Alan, Hayden Schaeffer, and Stanley Osher. "On the compressive spectral method." Multiscale Modeling & Simulation 12, no. 4 (2014): 1800-1827.

[Preprint]   [Paper]

40. Yang, Yi, Hayden Schaeffer, Wotao Yin, and Stanley Osher. "Mixing space-time derivatives for video compressive sensing." In 2013 Asilomar Conference on Signals, Systems and Computers, pp. 158-162. IEEE, 2013.

[Paper]

41. Schaeffer, Hayden, John Garnett, and Luminita Vese. "A texture model based on a concentration of measure." Inverse Problems & Imaging 7, no. 3 (2013).

[Paper]

42. Schaeffer, Hayden, Russel Caflisch, Cory D. Hauck, and Stanley Osher. "Sparse dynamics for partial differential equations." Proceedings of the National Academy of Sciences110, no. 17 (2013): 6634-6639.

[Preprint]   [Paper]

43. Duggan, Nóirín, Hayden Schaeffer, Carole Le Guyader, Edward Jones, Martin Glavin, and L. Vese. "Boundary detection in echocardiography using a Split Bregman edge detector and a topology preserving level set approach." In 2013 IEEE 10th International Symposium on Biomedical Imaging, pp. 73-76. IEEE, 2013.

[Paper]

44. Schaeffer, Hayden, and Stanley Osher. "A low patch-rank interpretation of texture." SIAM Journal on Imaging Sciences 6, no. 1 (2013): 226-262.

[Preprint]   [Paper]

Preprints:

1. Schaeffer, Hayden, Giang Tran, and Rachel Ward. "Learning dynamical systems and bifurcation via group sparsity." arXiv preprint arXiv:1709.01558 (2017).

[Preprint]   [Code]

2. Dimitrov, Evgeni, Hayden Schaeffer, David Wen, Sandra Rankovic, Kizza Nandyose, and Olivier Thonnard. "The Construction of a Vague Fuzzy Measure Through L Parameter Optimization." 2012

[Preprint]