Lennon J. Shikhman

Neural PDE solvers are increasingly used as learned surrogates, but in-distribution error alone does not show how they respond to structured shifts. This work stress-tests neural operators across dispersive, elliptic, fluid, financial, and chaotic PDE families, revealing failure patterns tied jointly to the architecture, equation, and type of distribution shift.

OpenReview 0S1LWZHQYn · arXiv:2601.11428

@article{shikhman2026diagnosing,
  title={Diagnosing Failure Modes of Neural Operators Across Diverse {PDE} Families},
  author={Shikhman, Lennon J.},
  journal={Transactions on Machine Learning Research},
  year={2026},
  url={https://openreview.net/forum?id=0S1LWZHQYn}
}

When boundary conditions vary, a neural PDE solver does not generally learn one boundary-agnostic solution operator. The learned map is indexed by the boundary distribution seen during training, so generalization in forcing terms or resolution does not imply generalization to new boundary conditions.

OpenReview lDjWQ9UxRy · arXiv:2603.01406

@inproceedings{shikhman2026one,
  title={One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural {PDE} Solvers},
  author={Shikhman, Lennon J.},
  booktitle={ICLR 2026 Workshop on AI and Partial Differential Equations},
  year={2026},
  url={https://openreview.net/forum?id=lDjWQ9UxRy}
}

This study asks whether new spacecraft inspection objectives can be added after launch without retraining a perception model. Frozen vision-language models use prompts to localize previously unseen spacecraft components, with strong performance on large structures and clear limitations on small appendages under orbital domain shift.

OpenReview w7pVSkgHRa · arXiv:2606.15427

@inproceedings{welsh2026postlaunch,
  title={Post-Launch Capability Expansion of Vision-Language Models via Prompting for On-Orbit Spacecraft Inspection},
  author={Welsh, Nicholas and Shikhman, Lennon J. and Attzs, Monty Nehru and Putane, Seemanthini Kusha and Nguyen, Van Minh and White, Ryan T.},
  booktitle={CVPR 2026 Workshop on AI for Space},
  year={2026},
  url={https://openreview.net/forum?id=w7pVSkgHRa}
}

Exact autonomous solution maps satisfy a semigroup law: evolving directly to time s + t agrees with evolving first to s and then by t. This paper turns that property into a post hoc diagnostic and shows that semigroup error is associated with rollout degradation in learned heat and Burgers simulators.

OpenReview MeAFOZnrvM · arXiv:2605.26324

@inproceedings{shikhman2026semigroup,
  title={Semigroup Consistency as a Diagnostic for Learned Physics Simulators},
  author={Shikhman, Lennon J.},
  booktitle={ICML 2026 Workshop on AI for Physics},
  year={2026},
  url={https://openreview.net/forum?id=MeAFOZnrvM}
}

Cellular Sheaf Neural Operators represent physical states on oriented cell complexes and couple local feature spaces through learned restriction maps. Incidence and Hodge structure place different quantities on appropriate cells and make selected compatibility constraints part of the update architecture.

arXiv:2606.00937

@article{shikhman2026cellular,
  title={Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained {PDE}s},
  author={Shikhman, Lennon J. and Gilbertie, Shane},
  journal={arXiv preprint arXiv:2606.00937},
  year={2026},
  url={https://arxiv.org/abs/2606.00937}
}

This work studies semilinear reaction-diffusion equations whose delayed feedback is represented by a finite signed Borel measure. It proves global weak well-posedness, robustness under total-variation and weak-star changes in the delay measure, and upper semicontinuity of compact global attractors under a delayed dissipativity condition.

arXiv:2606.04195

@article{shikhman2026kernel,
  title={Kernel-Robust Dynamics for Reaction-Diffusion Equations with Measure-Valued Delay},
  author={Shikhman, Lennon J.},
  journal={arXiv preprint arXiv:2606.04195},
  year={2026},
  url={https://arxiv.org/abs/2606.04195}
}

Operator Boosting constructs compact neural-operator surrogates through stagewise residual learning rather than training a large model and compressing it later. Across multiple architectures and PDE benchmarks, stacks of tiny operators frequently improve the empirical accuracy-parameter frontier while exposing regimes where boosting does not offset compression.

arXiv:2606.17460

@article{shikhman2026boosting,
  title={Operator Boosting Produces Pareto-Efficient {PDE} Surrogates},
  author={Shikhman, Lennon J.},
  journal={arXiv preprint arXiv:2606.17460},
  year={2026},
  url={https://arxiv.org/abs/2606.17460}
}

This software suite audits learned PDE simulators as approximate evolution operators rather than reducing evaluation to relative state error. It provides architecture-independent diagnostics for temporal composition, generator discrepancy, energy behavior, integral balance, admissibility, perturbation response, and scaling consistency.

arXiv:2606.18200

@article{shikhman2026diagnostic,
  title={A Diagnostic Software Suite for Auditing Learned {PDE} Simulators},
  author={Shikhman, Lennon J.},
  journal={arXiv preprint arXiv:2606.18200},
  year={2026},
  url={https://arxiv.org/abs/2606.18200}
}