Scientific machine learning
My research spans operator learning, inverse problems, uncertainty quantification, and differential equations, including neural and numerical methods, SDEs, functional differential equations (FDEs), and the analysis of PDEs.
01 / Research program
Structure-preserving operator learning
Inverse problems across scientific domains
Differential equations, structure, and uncertainty
R.01
Operator learning uses machine learning to approximate mappings between functions, such as taking an initial condition or forcing term to the solution of a differential equation. The goal is to build models that work across many inputs and can accelerate repeated scientific simulations.
Neural operators · geometric learning · multiphysics
R.02
Inverse problems use observed data to estimate unknown causes, parameters, states, or shapes. They arise when measurements are indirect, incomplete, or noisy and the quantities of interest cannot be observed directly.
Inverse methods · identifiability · data assimilation
R.03
Uncertainty quantification studies how uncertainty enters a model, affects its predictions, and should be communicated. It helps distinguish reliable conclusions from results that are sensitive to limited data, measurement noise, or modeling assumptions.
Function-space UQ · generative dynamics · consistency
R.04
Differential equations describe how systems change over time or space. This area includes stochastic differential equations (SDEs), functional differential equations (FDEs), and partial differential equations (PDEs), along with numerical, neural, and theoretical methods for studying their solutions.
SDEs · FDEs · PDEs
Live experiment / 01
Interactive map / 02
02 / Publications
Peer-reviewed publications and active research.
J.01
A cross-equation study of where neural-operator accuracy breaks and which diagnostics expose the failure.
Abstract
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.
Published record
W.01
Why changing boundary conditions changes the operator being learned, and what that means for generalization.
Abstract
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.
Published record
W.02
Expanding inspection capability after deployment through prompt-level adaptation rather than model retraining.
Abstract
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.
Published record
W.03
A model-agnostic test of whether learned time evolution composes like an autonomous physical flow.
Abstract
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.
Published record
P.01
A discretization-aware operator framework built on oriented cell complexes and learned restriction maps.
Abstract
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.
Published record
P.02
Well-posedness, robustness, and long-term behavior for reaction-diffusion systems with measure-valued memory.
Abstract
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.
Published record
P.03
Stagewise residual learning turns stacks of tiny operators into compact, frequently Pareto-improving PDE surrogates.
Abstract
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.
Published record
P.04
An architecture-independent panel for testing structural behavior beyond relative state error.
Abstract
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.
Published record
03 / Contact
I am especially interested in collaborations involving operator learning, inverse problems, uncertainty quantification, and differential equations, including SDEs, FDEs, neural and numerical methods, and the analysis of PDEs.
lj@shikhman.net