KHRONOS: Efficient Surrogate Modeling & SLAM

• KHRONOS is a computational framework that uses hierarchical kernel expansion and tensor decomposition for efficient surrogate modeling and spatio-temporal SLAM.
• It leverages a compact B-spline basis and CP tensor decomposition to achieve scalable, interpretable models with sub-millisecond inference and high accuracy.
• KHRONOS SLAM integrates local active window processing with global pose graph optimization to enable real-time, robust mapping in dynamic robotic environments.

KHRONOS is an acronym used for distinct, contemporary computational frameworks across several domains, notably scientific machine learning for surrogate modeling and spatio-temporal perception in robotics. The principal context in the literature emphasizes KHRONOS as a kernel-based neural architecture for rapid, resource-efficient scientific computation. In addition, Khronos has been introduced as a unified spatio-temporal metric-semantic SLAM solution. These frameworks leverage principled kernel expansion, tensor decomposition, and algorithmic factorization to achieve scalable, interpretable, and efficient learning and representation of high-dimensional physical systems and dynamic environments.

1. Mathematical and Architectural Foundations

KHRONOS (Kernel Expansion Hierarchy for Reduced Order, Neural Optimized Surrogates) addresses scientific computation involving high-dimensional fields via a hierarchical kernel expansion and tensorized mode superposition (Batley et al., 19 May 2025, Sarker et al., 11 Dec 2025). For an input $x \in \mathbb{R}^P$, a sequence of $L$ layers per coordinate $x_p$ is constructed as

$$f_p^{(l)}(x_p) = \sum_{i=1}^{N_{p,l}} w_{p,i}^{(l)}\, \mathcal{K}(f_p^{(l-1)}(x_p), \theta_{p,i}^{(l)}),$$

where $\mathcal{K}$ is a compactly-supported kernel, typically a quadratic B-spline. After $L$ layers, scalar-valued features $f_{p,j}^{(L)}(x_p)$ are obtained and tensorized into $J$ separable "modes": $M_j(x) = \prod_{p=1}^P f_{p,j}^{(L)}(x_p),$ with the surrogate field represented through a superposition

$\widehat{u}(x) = \sum_{j=1}^J M_j(x).$

This structure aligns with a CP (canonical polyadic) tensor decomposition, yielding linear scaling in both input dimension $P$ and rank $J$. The architecture is fully differentiable with all learning restricted to kernel knots and coefficient weights.

For applications in aerodynamic field prediction, KHRONOS additionally supports advanced multi-fidelity learning by composing a low-fidelity baseline and a ∆-correction module, each parametrized in the same kernel-tensor framework (Sarker et al., 11 Dec 2025). Model selection is performed via hyperparameter grid search (rank $r$, knot count $k$, learning rate, epochs), driven by the Galerkin functional in the native reproducing kernel Hilbert space (RKHS).

2. Loss Functions, Training Regimes, and Inference Complexity

Supervised learning is enabled by minimizing mean-squared error

$$L_{\text{mse}}(\theta) = \frac{1}{I}\sum_{i=1}^I \left[\widehat{u}(x_i;\theta) - u_i\right]^2,$$

but KHRONOS can be trained in a model-based (physics-informed) regime using strong and weak (Galerkin) residuals for target PDEs, or a mixed composite loss: $$L_{\text{mixed}} = \alpha_{\text{data}} L_{\text{mse}} + \alpha_{\text{model}} L_{\text{weak}}.$$ Optimization is unconstrained: stochastic first- or second-order optimizers (Adam, L–BFGS) are applicable, with batch sizes and learning rates problem-dependent.

Inference cost is dominated by inner-products: for $J$ modes and $P$ dimensions, the computational cost is $O(JP)$, enabling sub-millisecond field prediction even at high resolution. For inverse problems (e.g., level-set recovery), batch Gauss–Newton updates are implemented, yielding per-sample latencies in the sub-microsecond range on commodity GPUs. Compared to an MLP of width $W$ and depth $D$ (which scales as $O(DW^2)$), KHRONOS is orders of magnitude more efficient at fixed accuracy.

3. Benchmark Results and Empirical Scaling

On canonical benchmarks such as the 2D Poisson equation (domain $[0,1]^2$, Dirichlet boundary), KHRONOS attains $L_2^2$ errors from $4.7\times 10^{-4}$ (16 DoF) to $5.5\times 10^{-10}$ (512 DoF) with epoch times under $3$ ms (Batley et al., 19 May 2025). Observed scaling laws are

• $L_2^2 \sim$ DoF$^{-4}$,
• $H_1^2 \sim$ DoF$^{-3}$.

Relative accuracy is as follows:

• KHRONOS outperforms Kolmogorov-Arnold Networks by $100\times$ in error at fixed parameter count (~150), which already exceed MLP/PINN accuracy by $100\times$.
• Standard linear finite element methods (FEM) incur errors $10^4\times$ higher at comparable DoFs.

In multi-fidelity aerodynamic field prediction, KHRONOS achieves $R^2$ scores of up to $0.91$ with $2,500-17,900$ parameters (16–64 geometry points), compared to deep MLP/GNN/PINN counterparts requiring $1-6\times 10^5$ parameters (Sarker et al., 11 Dec 2025). Training times and inference latency are consistently $2-3\times$ faster for KHRONOS under tightly-bounded resource conditions. In regimes with limited high-fidelity (HF) data, KHRONOS offers significant gains in error convergence per parameter.

Method	Params (16 pts)	Inf. time (ms)	Train time (s)	$R^2$ (Case 3)
KHRONOS	7,759	3.27	8	0.90
MLP	139,554	5.49	40	0.87
GNN	202,626	7.84	65	0.92
PINN	139,554	5.52	48	0.90

4. Spatio-Temporal Perception: Khronos SLAM

Khronos (SLAM) is a unified approach for spatio-temporal metric-semantic SLAM in dynamic environments, targeting real-time dense 4D mapping on heterogeneous robotic platforms (Schmid et al., 2024). It formulates the perception problem as joint MAP estimation: $O^*, X^* = \arg\max_{O, X} P(O, X\,|\,Z, \Phi)$ where $O$ encodes all foreground/background objects, $X$ is the robot trajectory, $Z$ denotes semantic and geometric observations, and $\Phi$ are odometry increments. The system factors global state via locally-consistent "fragments" $Y_k$, supporting a two-tiered architecture:

• A local "active window" process fuses short-term dynamics at constant computational cost ($\sim22$ Hz).
• A global robust pose graph (via Truncated Least Squares) performs long-term reasoning, closing loops, and optimizing fragment/object assignment.

Fragment-matching operates via voxel IoU for merging measurements, and reconciliation is performed using a ray-library for robust change detection.

5. Data Structures, Implementation, and Computational Strategies

In KHRONOS surrogate modeling, each input coordinate is locally expanded in a quadratic B-spline basis, with tensor product composition enforcing separability. Model compression is inherent via CP decomposition and spline knot localization, yielding parameter-economical surrogates.

The SLAM variant utilizes:

• TSDF-based multi-resolution meshes for environment modeling,
• control-point graph deformation for the background,
• fragment pools for active/complete states, and
• hash-based structures for ray-wise reconciliation.

Pipeline implementations entail multi-threaded architecture with strict independence between real-time fusion/tracking and global optimization, ensuring robust performance spikes (e.g., TLS/PGO on loop closure) do not impede low-latency operation.

6. Practical Applications and Deployment Regimes

KHRONOS surrogate models have demonstrated applicability in:

• Online and embedded edge inference for control/surrogate evaluation,
• High-throughput inverse design and UQ,
• Real-time image-based surrogate modeling in computer vision,
• Multi-fidelity aerodynamic field estimation with strong resource constraints.

Khronos (SLAM) is validated on simulated (TESSE Apartment, Office) and real (Jackal, Spot) robotic datasets, achieving best-in-class F1 scores for background, static, dynamic objects, and scene change detection. In heterogeneous and large-scale environments, Khronos constructs accurate spatio-temporal maps at $\sim$20 Hz on CPUs.

7. Limitations and Open Directions

KHRONOS surrogates leverage explicit kernel-tensor architectures for parameter efficiency and built-in smoothness. Dense architectures may marginally outperform on large budgets or complex regimes, but only at $10-100\times$ higher computational cost (Sarker et al., 11 Dec 2025). All fragments are retained indefinitely in the SLAM context; marginalization of outdated elements is required for full lifelong mapping. Fragment association is currently translation-based, with scope for learned descriptors and robust matching under occlusion. Ray-based change detection depends on sufficient scene structure.

A plausible implication is that kernel-based tensor surrogates will remain central for efficient, scalable scientific machine learning under tight resource or latency constraints. The Khronos SLAM architecture suggests new approaches for integrating local and global reasoning in dynamic environments, modularizing spatio-temporal perception and extending to 4D dense mapping.

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References:

• "KHRONOS: a Kernel-Based Neural Architecture for Rapid, Resource-Efficient Scientific Computation" (Batley et al., 19 May 2025)
• "A Kernel-based Resource-efficient Neural Surrogate for Multi-fidelity Prediction of Aerodynamic Field" (Sarker et al., 11 Dec 2025)
• "Khronos: A Unified Approach for Spatio-Temporal Metric-Semantic SLAM in Dynamic Environments" (Schmid et al., 2024)