AAC Physics: Structure-Preserving Framework for Exact, Reversible Discrete-Time Convolutions

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AAC Physics: Structure-Preserving Framework for Exact, Reversible Discrete-Time Convolutions

Proprietary IP — Rich C. Young (Guru#1)

https://www.youtube.com/@GuruOne

Abstract

Memory-dependent discrete-time systems arise across physics, engineering, and ma- chine learning—from viscoelasticity and control to differentiable programming. Stan- dard recursive approaches accelerate convolution computation but suffer from drift, unstable backward passes, and numerical errors. AAC Physics introduces a structure-preserving, algebraically exact framework for discrete convolutions with exponential-like memory kernels. The method achieves:

• Exact equivalence to full discrete convolution (no drift),

• O(N) computational scaling with modest state storage,

• Algebraic reversibility, enabling machine-precision recovery of input sequences,

• Broad applicability across physics simulations, control systems, Neural ODEs, and gradient-based optimization.

1 Motivation and Background

Discrete convolutions with memory kernels appear in:

• Viscoelasticity: Stress–strain histories modeled via Prony-series exponentials.

• Control systems: Feedback with non-Markovian dynamics.

• Neural ODEs: Backpropagation through time with history-dependent states.

Challenges with existing methods:

• Naive summation is O(N 2) and infeasible for long sequences.

• Recursive approximations introduce drift and reduce forward/backward accuracy.

• Standard recursive methods are often non-invertible, limiting exact adjoint calcula- tions.

2 Core Concept: AAC Physics Framework

AAC Physics uses abstract, IP-protected functions to preserve structure and reversibil- ity:

• Forward update staten+1 = F(staten, inputn)

• Output computation outputn = G(staten, inputn)

• Backward recovery inputn = H(staten+1, staten)

• State step-back staten = I(staten+1, inputn)

Key properties:

1. Machine-precision equivalence to full discrete convolution.

2. Linear O(N) scaling for computation and memory.

3. Algebraic reversibility for exact backward reconstruction.

4. Proprietary coefficients protected within abstract functions F, G, H, I.

3 Performance and Accuracy

Table 1: Benchmark summary (IP-protected implementation).

Metric AAC Physics Standard Recursive Naive Convolution

Forward Error ∼10−16 O(∆t) drift Exact Backward Error ∼10−14 (quad) Divergent Exact Complexity O(N) O(N) O(N 2) Speedup ∼300× vs naive — —

4 Multi-Mode Exponential Kernel Architecture

Mode 2 Exponential kernel

Input sequence {vn}

Structure-preserving forward update

Mode 1 Exponential kernel

Output Backward rec

Mode 3 Exponential kernel

Figure 1: Multi-mode exponential kernel processing. Each kernel mode is updated indepen- dently and combined through a structure-preserving forward operator.

5 Forward and Backward Reversible Workflow

n = 0, . . . , N −1

Input {vn}

Forward update staten+1 = F(staten, vn)

Output computation yn = G(staten, vn)

Backward recovery vn = H(staten+1, staten)

State step-back staten = I(staten+1, vn)

Figure 2: Structure-preserving forward and backward workflow. Algebraic reversibility en- ables exact recovery of input samples without storing full state histories.

6 Abstract Pseudocode

Forward pass: for n = 0 .. N-1: state[n+1] = F(state[n], input[n]) output[n] = G(state[n], input[n])

Backward pass: for n = N-1 .. 0: input[n] = H(state[n+1], state[n]) state[n] = I(state[n+1], input[n])

7 Applications

• Physics simulations: viscoelasticity, fractional operators.

• Control and signal processing: non-Markovian feedback.

• Machine learning: Neural ODEs, adjoint computations.

• Industrial software: high-precision, memory-efficient simulations.

8 Summary

• O(N) efficient, drift-free, machine-precision forward computation.

• Algebraic reversibility enables exact backward reconstruction.

• Broad applicability across physics, control, and machine learning.

• Fully IP-protected while transparently demonstrating performance and accuracy.