Zero Coordinate Shift (ZCS): Operator Learning & Codes

• Zero Coordinate Shift (ZCS) is a set of mathematical techniques addressing both high-order differentiation in physics-informed operator learning and interference-free code sequence design in communications.
• In operator learning, ZCS collapses differentiation complexities by introducing a shared shift variable, dramatically reducing computational memory and execution time.
• In communications, ZCS underpins zero correlation zone (ZCZ) sequence design, ensuring multi-user signal transmissions remain robust and interference-free.

Zero Coordinate Shift (ZCS) refers to a class of algorithmic and mathematical constructions that, depending on research context, address either (1) optimally structured code sequences for communication with guaranteed zero correlation within specific shift zones, or (2) efficient automatic differentiation for operator learning in physics-informed deep learning. Despite sharing a name, these usages are distinct in scope and technical realization. The following entry details both interpretations as independently developed in the contemporary literature.

1. ZCS in Physics-Informed Operator Learning

Motivation and Background

Zero Coordinate Shift (ZCS) in the context of physics-informed operator learning addresses the scalability bottleneck in automatic differentiation (AD) when computing high-order spatial or temporal derivatives with respect to collocation coordinates in Physics-Informed Neural Operators (PINOs) such as DeepONets. Standard AD frameworks require making all sampled coordinates leaf nodes, resulting in a many-roots-many-leaves (_∞∞) problem; this approach causes computational and memory requirements to scale poorly with the number of input functions $M$ and collocation points $N$ (Leng et al., 2023).

Existing attempts to circumvent this include explicit function loops (computing separate graphs for each input function) and data vectorization (treating the entire $(M \times N)$ field as a single vector), each incurring $O(M)$ memory and compute blowup. ZCS introduces a shared infinitesimal shift variable $z$ (per spatial dimension), collapsing these differentiation complexities into substantially smaller computational graphs.

Mathematical Formulation

Let $u_{ij} = f_\phi(p_i, x_j)$ denote the network output for function $p_i$ at collocation point $x_j$. Standard AD computes derivatives $\partial u_{ij}/\partial x_j$ by making all coordinates $\{x_j\}$ leaves in the graph. ZCS introduces a scalar $N$0 so that

$N$1

which enables the desired derivative to be obtained as

$N$2

Here, only $N$3 is a leaf, shifting all reliance on the coordinate dimension to a single computational node.

To recover the full $N$4 tensor of derivatives efficiently, one introduces a root-collapsing weight tensor $N$5 and considers

$N$6

The entries $N$7 are then available via $N$8, and the corresponding derivatives as mixed second derivatives $N$9. Higher-order derivatives and mixed products follow recursively by this formalism, reducing computation from an _∞∞ problem to a sequence of _11 and _∞1 operations suitable for reverse-mode AD.

Algorithmic Realization

The procedure, expressed in PyTorch pseudocode for two spatial dimensions $(M \times N)$0, is as follows:

$x_j$7

This method requires only three scalar-to-scalar (11) and one vector-to-scalar (∞1) AD operations, eliminating the need for loops or coordinate duplication.

Implementation Considerations

The ZCS methodology is implemented by subclassing DeepXDE’s operator modules, overriding coordinate handling to use shift scalars and compressing the AD graph. The approach treats $(M \times N)$1 as a batch dimension, with coordinate tensors never duplicated across input functions. The bulk memory and compute thus remain comparable to $(M \times N)$2 PINN instances, with the only overhead coming from a small number of scalars and possibly the $(M \times N)$3 tensor.

Empirical Performance

Extensive benchmarking on representative PDE problems (reaction-diffusion, Burgers, Kirchhoff–Love plates, Stokes) demonstrates that ZCS reduces computational graph memory consumption and wall time by an order of magnitude compared to previous approaches. For example, in a Burgers equation with $(M \times N)$4, ZCS achieves 0.20 GB graph memory, 0.36 GB peak memory, and 15 s per 1000 batches, versus 7.8 GB, 7.9 GB, and 316 s for function loop methods (Leng et al., 2023). Relative errors are essentially unchanged between methods.

A summary table is provided:

Problem (M,N,P)	Method	Graph Mem (GB)	Peak Mem (GB)	Time/1000 batches (s)	Rel. Error
Reaction–Diff (50,1000,2)	FuncLoop	0.96	0.98	181	8.3% ±2.0
	DataVect	0.97	1.46	24	9.5% ±2.5
	ZCS	0.02	0.05	10	8.2% ±2.0
Burgers (50,12800,2)	FuncLoop	7.84	7.91	316	7.5%
	DataVect	7.73	11.40	221	7.2%
	ZCS	0.20	0.36	15	7.1% ±0.5
Kirchhoff–Love (36,10000,4)	FuncLoop	77.6	77.6	4081	27.3%
	DataVect	–	–	–	–
	ZCS	2.36	3.30	144	26.9% ±0.5
Stokes (50,5000,2)	FuncLoop	74.4	78.6	4253	10.3%
	DataVect	–	–	–	–
	ZCS	1.99	3.30	147	10.4% ±0.6

A plausible implication is that ZCS facilitates the training of operator learning models at previously infeasible scales.

Scope and Limitations

ZCS is agnostic to the choice of network architecture, PDE type, or data sampling, and imposes no loss in training accuracy. It does not yield advantages for architectures leveraging explicit grid structure (e.g., Fourier Neural Operators or CNN-based PINNs) with optimized finite difference or FFT differentiators, where $(M \times N)$5 stencils remain optimal. ZCS is formulated for reverse-mode AD, and its application to forward-mode scenarios is an area for potential extension (Leng et al., 2023).

2. ZCS and Zero Correlation Zone (ZCZ) Sequences

Definition and Theoretical Foundations

In communication theory, Zero Coordinate Shift (ZCS) is closely related to the zero correlation zone (ZCZ) property of sequences used for multi-carrier code division multiple access (MC-CDMA) and related systems. For two complex-valued sequences $(M \times N)$6 and $(M \times N)$7, the aperiodic cross-correlation is defined as

$(M \times N)$8

ZCZ of width $(M \times N)$9 means $O(M)$0 for all nonzero $O(M)$1. For a collection of sequences, when auto- and cross-correlations satisfy this property, the set is a Z-complementary code set (ZCCS) (Shen et al., 2021).

A $O(M)$2–ZCCS is a family $O(M)$3 of $O(M)$4 sequence sets, each $O(M)$5 consisting of $O(M)$6 sequences of length $O(M)$7. The ZCCS definition requires

1. $O(M)$8 for $O(M)$9 (sum auto-correlation vanishes outside zero shift).
2. $z$0 for all $z$1 and $z$2 (mutual orthogonality over the zone).

The Feng–Fan–Zhou bound asserts $z$3; equality means optimality.

Construction via Extended Boolean Functions

Optimal ZCCS constructions are achieved using extended Boolean functions (EBF) $z$4:

$z$5

with $z$6, $z$7, $z$8, $z$9 a permutation of $u_{ij} = f_\phi(p_i, x_j)$0, $u_{ij} = f_\phi(p_i, x_j)$1, $u_{ij} = f_\phi(p_i, x_j)$2.

Define

$u_{ij} = f_\phi(p_i, x_j)$3

for $u_{ij} = f_\phi(p_i, x_j)$4 (in base $u_{ij} = f_\phi(p_i, x_j)$5), $u_{ij} = f_\phi(p_i, x_j)$6.

Partitioning the $u_{ij} = f_\phi(p_i, x_j)$7 sequences into $u_{ij} = f_\phi(p_i, x_j)$8 sets $u_{ij} = f_\phi(p_i, x_j)$9 yields an optimal $p_i$0–ZCCS: correlation vanishes up to shifts $p_i$1 (proof by generalized Golay argument and monomial analysis) (Shen et al., 2021).

A concrete illustration: for $p_i$2, $p_i$3, $p_i$4, $p_i$5, one recovers a $p_i$6–ZCCS by explicit computation.

Practical Implications

ZCZ sequences constructed this way are critical for asynchronous MC-CDMA, enabling multiple users to transmit signals robustly with delay offsets up to $p_i$7 without multi-user or multi-path interference. The practical trade-off—$p_i$8—permits system design flexibility by adjusting sequence length $p_i$9, flock size $x_j$0, and zone width $x_j$1. The $x_j$2-ary construction generalizes beyond binary codes, supporting large, optimal ZCCS for high-capacity networks (Shen et al., 2021).

3. Applications in Deep Learning and Communications

In physics-informed deep learning, ZCS enables efficient training of neural operators for PDEs, with empirical evidence indicating scalability to previously unreachable sizes for operator dimensions $x_j$3 and collocation points $x_j$4, without increased training errors (Leng et al., 2023).

In wireless communications, ZCS in the ZCCS/ZCZ sense is foundational to MC-CDMA and analogous systems, supporting robust, interference-free multi-user transmission and flexible protocol design (Shen et al., 2021).

4. Comparative Summary and Terminological Distinctions

ZCS, as developed independently in operator learning and communications code design, addresses orthogonal concerns: backpropagation efficiency in the former, and interference-robust sequence structure in the latter. In operator learning, ZCS is a differentiable shift variable trick; in sequence design, it is a condition on correlation functions. Each is grounded in rigorous mathematical analysis and has led to substantial advances in its respective field.

5. Limitations and Prospective Extensions

In physics-informed operator learning, ZCS is not advantageous for grid-based models amenable to $x_j$5 stencil or FFT-based differentiation, and is primarily realized for reverse-mode AD systems. Anticipated extensions include porting to forward-mode AD in frameworks such as JAX or Julia for further performance improvement at high derivative orders.

In code design, ZCS/ZCZ constructions hinge on the algebraic constraints of the employed EBFs, and while highly flexible in $x_j$6-ary settings, remain subject to combinatorial existence bounds (Feng–Fan–Zhou).

• "Zero Coordinate Shift: Whetted Automatic Differentiation for Physics-informed Operator Learning" (Leng et al., 2023)
• "New Construction of Z-Complementary Code Sets and Mutually Orthogonal Complementary Sequence Sets" (Shen et al., 2021)