Lab for Parallel Numerical Algorithms
We are developing a Markov chain Monte Carlo algorithm to estimate contractions of closed tensor networks. This allows efficient approximation of quantities like
\[
\text{Tr}(ABCD) = \sum_{ijkl} A_{ij}B_{jk}C_{kl}D_{li}
\]
using methods from statistical physics to improve mixing.
Such contractions are trivial in small cases but become \(\#\textsf{P}\)-hard for large networks. Our approach targets use cases in quantum circuits and chemistry, where exact contraction is intractable.
We'll present this work at QSim 2025. Read more here!
Computation & Neurodynamics Lab
I study the dynamics of stochastic neuronal populations using moment closure methods. Starting from nonlinear SDEs that describe individual neuron behavior, we analytically derive reduced ODEs governing the evolution of the population mean and covariance. This requires evaluating Jacobians, stochastic forcing terms, and closure approximations to capture the effect of higher-order moments.
The system under consideration takes the form \[ dx = f(x, \lambda) \, dt + G(x, \lambda) \, dW_t \] where \(x(t)\) is the neuronal state (e.g., membrane voltage and recovery variable), \(\lambda\) encodes input heterogeneity across the population, and \(W_t\) is vector-valued white noise. Nonlinearities in \(f\) (e.g., cubic terms or feedback) make exact inference intractable, so we evolve only the first two moments, which often suffice to capture collective dynamics.
The closure equations are symbolically derived using sympy and validated against Monte Carlo simulations. We’re now analyzing these ODEs through the lens of symmetry and geometric structure: computing Lie brackets between the dynamics and candidate vector fields to uncover invariants, symmetry-preserving perturbations, and coordinate-free signatures of neural computation.
Current experiments include building surrogate models that learn the map \((\lambda, t) \mapsto (\mu(t), \Sigma(t))\), benchmarking architectures (e.g., neural nets vs. GPR), and extending the framework to rich biophysical models such as FitzHugh–Nagumo. Full implementation is on GitHub.
More updates soon!