About
I build numerical solvers and computational systems from first principles.
My work sits at the intersection of computational mathematics, geometric mechanics, and scientific machine learning. Rather than treating models as black boxes, I design computational architectures that respect the underlying mathematical structure—forcing algorithms to preserve geometric invariants, phase-space structure, and physical conservation laws.
When approaching complex or nonlinear dynamical systems, I look at the failure modes and extreme limits first. That is usually where naive approximations break down, where numerical diffusion creeps in, and where the true governing geometry reveals itself. Whether working on fluid turbulence, chaotic multi-body dynamics, high-dimensional stochastic PDEs, or foundation time-series architectures, the objective remains constant: building methods that hold up when conditions become extreme.
Outside of mathematical derivations and custom PyTorch/C++ routines, intellectual work cannot be separated from physical reality. Working with tractors and agriculture in my village, maintaining and driving machinery to understand mechanical thresholds, and daily heavy physical training keep my thinking grounded. Freehand pencil sketching trained my eye to understand continuous curvature, perspective, and spatial balance long before formalizing them into differential geometry.
The standard is straightforward: things must work under load—whether it is a mechanical engine, an athletic lift, or an algorithm. Everything I formulate is developed in the open with publicly archived code, manuscripts, and reproduction pipelines.
Flagship Research & Systems
Working Paper • Target: NeurIPS / SIMODS
Contact-SSM: Geometric Contact State-Space Models for Time Series
Embeds dynamical latent states onto an odd-dimensional contact manifold $\mathbb{R}^{2d+1}$ with contact 1-form $\alpha = \mathrm{d}s - \mathbf{p}^T\mathrm{d}\mathbf{q}$ and Reeb vector field $R = \partial/\partial s$, modeling dissipative and driven dynamical systems without secular phase drift. Integrates the infinitesimal generator $K \in \mathfrak{sp}(2d) \oplus \mathbb{R}$ via Cayley transform into parallel associative prefix scans, achieving $\mathcal{O}(L)$ GPU training and $\mathcal{O}(1)$ real-time streaming rollouts without the quadratic overhead of Transformers.
$\alpha = \mathrm{d}s - \mathbf{p}^T\mathrm{d}\mathbf{q}, \quad \iota_{X_{\mathcal{H}}} \alpha = -\mathcal{H}, \quad \iota_{X_{\mathcal{H}}} \mathrm{d}\alpha = \mathrm{d}\mathcal{H} - (R \mathcal{H})\alpha$
Under Review • Journal of Computational Physics (Elsevier)
Parameter-Space Error Localization and Posterior Fidelity of PINN Surrogates
Comprehensive 59-page theoretical and empirical investigation demonstrating that surrogate approximation errors enter Bayesian posteriors strictly through sparse sensor projections and localized likelihood filtering. Proved inverse-noise Total Variation and Wasserstein-1 stability bounds $\mathcal{O}(\sigma_{\mathrm{noise}}^{-2})$, formulated the Bayesian Fidelity Ratio (BFR) screening threshold across a balanced 240-model ensemble, and uncovered an observational projection crossover in 2D Incompressible Navier-Stokes and 5D multi-mode diffusion.
$\mathcal{W}_1(\pi, \widehat{\pi}) \le \mathrm{diam}(\Theta) \cdot d_{\mathrm{TV}}(\pi, \widehat{\pi}) \le \frac{\mathrm{diam}(\Theta)}{2}\left[\exp(2R\varepsilon + \varepsilon^2) - 1\right] = \mathcal{O}(\sigma_{\mathrm{noise}}^{-2})$
Under Review • IEEE Transactions on Computational Imaging
Derivative-Informed Neural Forward Surrogates for EIT Shape Inversion
Resolves the severe ill-posedness of the continuous Calderón problem in Electrical Impedance Tomography (EIT). Formulates a 64-dimensional periodic cubic B-spline with stochastic directional Jacobian-Vector Product (JVP) supervision on $\mathbb{S}^{63}$, reducing retained autograd branches from 64 to 1 per sample. Replaces online FEM solves entirely ($N_{\mathrm{FEM}}^{\mathrm{online}} = 0$) to achieve a $56.8\times$ wall-clock speedup ($35.6\,\mathrm{ms}$ vs $2.02\,\mathrm{s}$/step) with bounded $342.8\,\mathrm{MB}$ VRAM and $99.57\%$ noise robustness across a 234-trial benchmark.
$\mathbf{J}_{\mathrm{FEM}} = \mathbf{J}_\sigma \frac{\partial \boldsymbol{\Sigma}}{\partial \boldsymbol{\theta}}, \quad \mathbb{E}_{\mathbf{v} \sim \mathbb{S}^{63}} \left[ \|\mathbf{J}_{\mathrm{surr}}\mathbf{v} - \mathbf{J}_{\mathrm{FEM}}\mathbf{v}\|_2^2 \right] \propto \|\mathbf{J}_{\mathrm{surr}} - \mathbf{J}_{\mathrm{FEM}}\|_F^2$
Under Review • Physics of Fluids (AIP Publishing) [MS #POF26-AR-14856]
Operator Conditioning and False Convergence in Incompressible PINNs
Controlled regularized lid-driven cavity study ($Re=1000$). Proved that continuous mesh-free PINNs lack discrete stencils for Thom's wall-vorticity condition, causing False Convergence and Operator Diffusion in streamfunction-vorticity ($\psi$--$\omega$) models ($67.3\times$ residual drop while underpredicting secondary corner eddies by $14.3\%$). Proved that retaining pressure $p$ in $\psi$--$p$ enforces an implicit elliptic Helmholtz–Hodge projection, restoring operator conditioning ($14.2\%$ vs $2.8\%$ global velocity $L_2$ error).
Under Review • Applied Mathematical Modelling (Elsevier)
High-Resolution Finite-Difference Framework up to Re = 100,000
Engineered an exact $\mathcal{O}(N^2 \log N)$ Discrete Sine Transform (DST-I) Poisson engine with vectorized ADI marching on $1025 \times 1025$ grids ($1.05\mathrm{M}$ nodes), eliminating iterative Poisson drift. Preserved pure central differencing ($\nu_{\mathrm{art}} = 0$) up to $Re = 50,000$, verified Batchelor's core theorem and ASME V&V 20 Richardson GCI ($0.48\%$), and resolved unsteady periodic vortex shedding ($St = 0.6249$) up to $Re = 100,000$.
Working Paper • Target: MNRAS / Astronomy & Computing
CHRONOS: Canonical Hamiltonian Reconstruction via Symplectic Potential Inversion
Joint work with Prof. Vinay Kumar. Proved Theorem 1 (Separable Kinetic-Coriolis Decomposition), Theorem 2 (Arnold Extended Phase Space $\mathbb{R}^{2d+2}$), and Theorem 3 (Separable Extended Synthesis), preserving Jacobi invariants ($|\Delta\mathcal{I}| < 10^{-1}$) across 7 non-integrable celestial systems. Discovered the Far-Field Oscillatory Ringing Trap and eliminated ripples via Mexican Hat Wavelet Nets ($2.0 \times 10^{11}\times$ cleaner curvature). Built Sundman symplectic steppers resolving KAM tori with >1,100 Poincaré punctures.
Working Paper • Target: SIAM J. Scientific Computing (SISC)
AS-PINN: Autonomous Parameter-Space Adaptive Mesh Refinement
Engineered autonomous domain decomposition via vectorized per-sample Gram alignment profiling ($\mathcal{G}_{ij} = \cos \angle(\mathbf{g}_i, \mathbf{g}_j)$ via torch.func.vmap) with exact zero-disruption cleavage invariance ($\|u^{(N+1)} - u^{(N)}\| = 0$). Attained a $725.6\times$ loss reduction over standard PINNs, PCGrad, and CAGrad on high-frequency Helmholtz ($k=4\pi$) and singular 2D Klein-Gordon systems.
Working Paper • Target: IEEE TPAMI / CVPR
AS-ViT: Dynamic Latent Expert Routing in Multi-Task Vision
Resolved multi-task negative transfer in dense vision backbones by tracking inter-task Gram matrix negative eigenvalues ($\lambda_{\min}(\mathcal{G}) < -\tau$). Dynamically cleaves dedicated latent expert subspaces using Partition of Unity (PoU) gating, delivering a $+5.84\%$ mean multi-task gain across semantic segmentation, monocular depth estimation, and 3D surface normal prediction on the NYUv2 benchmark.
Working Paper • Target: Journal of Computational Physics (JCP)
Null-Space PINN: Algebraic Gradient Orthogonality on Boundary Domains
Decomposed the neural parameter manifold into orthogonal direct-sum subspaces ($\Theta = \Theta_0 \oplus \Theta_1$, $\mathcal{W}_0 \mathcal{W}_1^\top = \mathbf{0}$) blended via a $C^2$ Quintic Hermite seam operator ($\psi(\xi) = 6\xi^5 - 15\xi^4 + 10\xi^3$). Proved structural gradient orthogonality ($\langle \nabla \mathcal{L}_{\mathrm{if}}, \nabla \mathcal{L}_{\mathrm{des}} \rangle \equiv 0$) on saturated boundary domains, eliminating gradient pathology in stiff reaction-diffusion equations.
Research Engine • Quantitative Finance
Deep-EEP-PINN: Pricing High-Dimensional American Basket Options
Priced correlated American basket options up to $d = 50$ underlying assets (1,225 correlations), breaking Bellman's Curse of Dimensionality ($N^d$) via Directional Autograd Hessian Trace Contraction in $\mathcal{O}(d)$ linear complexity ($< 3\,\mathrm{GB}$ VRAM). Validated against 100,000-path Longstaff–Schwartz Monte Carlo and Crank–Nicolson PSOR with $0.37\%$ relative pricing discrepancy.
Research Engine • Uncertainty Quantification
PINN-Bayesian-Posterior-Fidelity: Surrogate Distortion Diagnostics
Diagnostic framework quantifying neural surrogate-induced Bayesian posterior distortion in inverse problem characterization. Formulates the 1-Wasserstein Bayesian Fidelity Ratio ($\mathrm{BFR}$) normalized by empirical MCMC stochastic noise floors, isolating exact spatial regions where PINN approximation errors corrupt epistemic parameter uncertainty.
Open-Source Framework • Target: JOSS
BRSDK: Real-Time In-Engine Physics Telemetry for Autonomous Vehicles
High-frequency telemetry extraction framework operating inside BeamNG's 2,000\,Hz soft-body vehicle physics thread with zero dynamic Lua heap allocations during logging. Built an Apache Arrow columnar Python SDK with Polars streaming and zero-copy DLPack PyTorch tensor interchange for offline reinforcement learning and state estimation.
