JensUn Approach Framework

• The approach redefines interval arithmetic by completing its algebraic structure into a Banach space for enhanced stability in computations.
• It embeds intervals into a four-dimensional associative algebra, enabling efficient and reliable symbolic and numeric algorithms.
• It integrates sensing-aided filtering and neural network emulation to enforce physical sensitivities and improve estimation performance in uncertain systems.

The JensUn Approach refers to a methodologically rigorous framework for computation and estimation in settings with uncertainty, especially where classical interval arithmetic, robust communication-sensing systems, or machine learning emulators for dynamical systems are required to provide reliable, physically consistent outputs. Across its development and application, the JensUn Approach leverages additional algebraic, analytic, and statistical structures to achieve consistency, accuracy, and computational tractability in uncertain environments. Its methodology is manifest in the augmentation of interval arithmetic using Banach and associative algebraic structures, the enhancement of channel estimation in joint communication and sensing via sensing-aided filtering, and the explicit enforcement of physical sensitivities in neural network emulators. Below, the principal components of the JensUn Approach are discussed across representative settings.

1. Algebraic Formalism for Interval Arithmetic

The JensUn Approach introduces a completed algebraic structure for intervals, transforming their classical semiring into an abelian group and vector space. Any closed interval $[a, b]$ is symmetrized via elements $\overline{(A, 0)}$ or $\overline{(0, A)}$, allowing for externally defined scalar multiplication:

$$\alpha \cdot \overline{(A, 0)} = \overline{(\alpha A, 0)} \quad (\alpha > 0),$$

with sign-flip accommodation for $\alpha < 0$. The ensuing space $\overline{\mathbb{IR}}$ admits a norm:

$$\| \mathcal{X} \| = l(\mathcal{X}) + |c(\mathcal{X})|, \qquad l(X) = b-a,\, c(X) = \frac{a+b}{2},$$

where $\mathcal{X}$ is associated with interval $A=[a,b]$. The resulting Banach space underpins rigorous operations—including limit processes and continuity—on interval-valued representations. This regularization is essential for stable iterative algorithms and analytic extension in imprecise computational domains (0809.5173).

2. Associative Algebraic Embedding and Linearization of Interval Operations

A distinctive aspect of the JensUn Approach is the embedding of intervals into a four-dimensional associative real algebra $\mathcal{A}_4$ defined on the canonical basis $\{e_1, e_2, e_3, e_4\}$, where the multiplication table is given by:

$$x \cdot y = (x_1y_1 + x_4y_4,\; x_2y_2 + x_3y_3,\; x_3y_2 + x_2y_3,\; x_4y_1 + x_1y_4),$$

for $$x = (x_1, x_2, x_3, x_4),\, y = (y_1, y_2, y_3, y_4) \in \mathcal{A}_4$$, with unit $e_1 + e_2$. Intervals $[x_1, x_2]$ are embedded by mapping:

• $(x_1, x_2, 0, 0)$ if $x_1, x_2 \geq 0$,
• $(0, x_2, -x_1, 0)$ if $x_1 \leq 0 \leq x_2$,
• $(0, 0, -x_1, -x_2)$ if $x_1, x_2 \leq 0$.

This embedding ensures classical interval product corresponds to algebraic multiplication in $\mathcal{A}_4$. With associative and distributive structure, symbolic and numeric algorithms can be implemented efficiently, and interval operations can be systematically projected back to interval domains—facilitating enclosed and reliable solutions (0809.5173).

3. Interval Divisibility and Euclidean-Type Division

Beyond invertibility, the JensUn Approach analyzes divisibility and extends Euclidean division algorithms to interval spaces. An interval in $\mathcal{A}_4$ is invertible iff $(x_1^2 - x_4^2)(x_2^2 - x_3^2) \ne 0$. For positive intervals $\mathcal{X} = \overline{([x_1, x_2], 0)}$, division proceeds via $\mathcal{Z}= \mathcal{X}^{-1}\cdot \mathcal{Y}$; for non-invertible cases, given ordering on endpoint ratios, a unique pair $(\mathcal{Z}, \mathcal{R})$ exists satisfying $$\mathcal{Y} = \mathcal{X}\cdot \mathcal{Z} + \mathcal{R}$$, with $l(\mathcal{R}) = 0$ and minimal center:

$$c(\mathcal{Z}) = \frac{1}{2} \left( \frac{y_2}{x_2} + \frac{y_1}{x_1} \right), \quad l(\mathcal{Z}) = \frac{y_2}{x_2} - \frac{y_1}{x_1}$$

This construction provides robust algebraic tools for equation solving and is critical for stability in iterative numerical processes where uncertainties must be precisely bounded (0809.5173).

4. Extension to Differential Calculus in Interval Spaces

Endowing the interval setting with a Banach space structure facilitates the rigorous deployment of differential calculus. Differentiability for functions $$f : \overline{\mathbb{IR}} \to \overline{\mathbb{R}}$$ is defined: there exists a linear map $L$ such that:

$$\| f(\mathcal{X}) - f(\mathcal{X}_0) - L(\mathcal{X} - \mathcal{X}_0) \| = o(\| \mathcal{X} - \mathcal{X}_0 \|)\text{ as } \mathcal{X} \to \mathcal{X}_0.$$

Analysis of functions such as $q_2(\overline{([a, b], 0)})$, which produces interval squares (with cases for $a<b<0$ and $a<0<b$), demonstrates that continuous interval functions may not be differentiable throughout their domain, marking the need for tailored approaches to interval-valued function analysis. This extension allows the adaptation of gradient-based and variational optimization methods in uncertain domains (0809.5173).

5. Sensing-Aided Estimation in Joint Communication and Sensing Systems

In real-time communication scenarios, the JensUn Approach is echoed in the SAKF-based channel state information (CSI) estimator for joint communication and sensing (JCAS). Initial CSI is estimated via least squares (LS), then refined sequentially using a Kalman filter along antenna array axes—leveraging angle-of-arrival (AoA) priors:

$$\hat{A}_P = \exp(-j(2\pi d_a/\lambda)\cos\widehat{\varphi}\sin\widehat{\theta}), \quad \hat{A}_Q = \exp(-j(2\pi d_a/\lambda)\sin\widehat{\varphi}\sin\widehat{\theta})$$

The Kalman filter predicts and updates in both array axes, recursively reducing estimation noise. This approach achieves a reduction in required SNR of about 1.8 dB to match LS performance and approaches MMSE accuracy (within 0.2 dB), while complexity drops from $O(N^3)$ in MMSE to $O(3N)$ in SAKF. The reuse of sensing results in communication estimation typifies the cross-domain rigor of the JensUn Approach (Chen et al., 2022).

6. Enforcement of Physical Sensitivities in ML Dynamical System Emulators

The JensUn Approach is further exemplified in the JENN framework for neural network dynamical model emulation. Here, a two-phase training regime first establishes forecast capability and later enforces tangent linear (TL) and adjoint (AD) consistency—directly penalizing RMSE in TL/AD emulation:

$$L_{total} = \alpha L_{forecast} + \beta L_{TLM} + \gamma L_{ADJ},$$

where $L_{forecast}$ is RMSE on true/predicted state, $L_{TLM}$ and $L_{ADJ}$ are sensitivity RMSEs. This enables the network to emulate both primary nonlinear dynamics and sensitivity structures, essential for data assimilation. Notably, Jacobian enforcement does not require model retraining from scratch or architectural change, making the approach extensible to pretrained models such as GraphCast, Pangu, NeuralGCM, and FuXi. Experiments on the Lorenz96 model show forecast accuracy is preserved while TL/AD oscillations and Jacobian noise are suppressed—directly benefiting operational assimilation tasks (Tian, 2 Dec 2024).

7. Applications and Impact on Computational Methodologies

The rigorous structure and analytic capabilities introduced in the JensUn Approach substantially enhance classical and machine learning-based estimation methodologies in imprecise domains. In interval mathematics, linear and nonlinear programming, error estimation, and optimization become amenable to robust guarantees via algebraic embedding and differential extension. In advanced communications, sensing-aided filtering enables efficient, accurate real-time operation under spectrum constraints. In ML-driven forecasting, explicit Jacobian enforcement closes the gap between physical and data-driven emulators, enabling reliable data assimilation in operational settings.

The JensUn Approach thus underpins a coherent paradigm for structuring computations in uncertain, multi-domain environments, supporting both theoretical advances and practical robustness in scientific and engineering applications. Its adoption in interval arithmetic, JCAS estimation, and data assimilation governs not only the fidelity of numerical outcomes but also their operational tractability in high-performance, uncertainty-laden computational systems.