Deceptron Module: Accelerating Inverse Solvers

• Deceptron module is a bidirectional learned local-inverse architecture that pairs forward and reverse neural surrogates to efficiently address nonlinear inverse problems.
• It employs explicit left-inverse regularization and a Jacobian Composition Penalty to enforce local geometric consistency and enable Gauss–Newton–like updates with low computational overhead.
• Empirical results show its D-IPG solver rapidly converges with significantly fewer iterations and reduced cost compared to traditional gradient-based or second-order methods.

The Deceptron module is a bidirectional learned local-inverse architecture designed to accelerate and stabilize nonlinear inverse problems in physics and engineering by amortizing local inverse geometry. It comprises paired forward and reverse neural surrogates with explicit left-inverse regularization, enabling Gauss–Newton–like updates at inference with lightweight computational overhead. The Deceptron and its iterative solver, D-IPG (Deceptron Inverse-Preconditioned Gradient), provide a practical alternative to traditional gradient-based and second-order optimization methods by learning and utilizing a reusable, data-driven local inverse map (Kachhadiya, 26 Nov 2025, Kachhadiya, 13 May 2026).

1. Mathematical Structure and Module Definition

The Deceptron module operates on the general nonlinear least-squares inverse problem:

$$\min_{x \in \mathbb{R}^{d_\text{in}}} \ \Phi(x) = \frac{1}{2}\Vert f_W(x) - y^\star \Vert_2^2,$$

where $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$ is a differentiable “forward surrogate” parameterized by $W$ (e.g., neural network weights), and $y^\star$ is a measured or desired output. The Deceptron introduces a paired reverse map $$g_V\!: \mathbb{R}^{d_\text{out}} \to \mathbb{R}^{d_\text{in}}$$ parameterized by $V$. This bidirectional structure does not target a global inverse but learns $g_V$ as a local left-inverse for $f_W$ in the optimization region of interest:

$\nabla_y g_V(y) \approx [\nabla_x f_W(x)]^+,$

where $^+$ denotes the Moore–Penrose pseudoinverse.

Architecturally, $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$0 and $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$1 are typically shallow NNs (MLPs or residual CNNs) matched to the problem structure. Bias parameters $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$2 (forward) and $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$3 (inverse) are softly tied by a penalty $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$4 to encourage local invertibility near zero (Kachhadiya, 26 Nov 2025).

2. Training Objectives and Jacobian Composition Penalty

Deceptron modules are trained with a composite objective to incentivize forward-surrogate fitness, local inversion, and Jacobian-based geometry:

• Supervised fit: $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$5
• Reverse reconstruction: $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$6
• Cycle consistency: $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$7
• Spectral regularization: $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$8
• Soft bias tie: $$f_W\!: \mathbb{R}^{d_\text{in}} \to \mathbb{R}^{d_\text{out}}$$9
• Jacobian Composition Penalty (JCP): Enforces local left-inverse structure via

$W$0

where $W$1 are Jacobians of $W$2 and $W$3, and $W$4 is a random probe (Rademacher or Gaussian) enabling efficient computation via Hutchinson’s estimator.

The total training loss is a sum of these terms, controlling the output-space fidelity, bidirectional invertibility, and geometric left-inverse quality (Kachhadiya, 26 Nov 2025, Kachhadiya, 13 May 2026).

3. D-IPG Inverse-Preconditioned Solver

At inference, the Deceptron is deployed inside D-IPG, which implements an inverse-preconditioned gradient update:

1. Compute forward: $W$5, residual $W$6.
2. Output-space descent: $W$7.
3. Pullback: $W$8.
4. Relaxed projection: $W$9.
5. Accept trial point if Armijo condition is met; otherwise backtrack.

To first order,

$y^\star$0

and if $y^\star$1, the step matches damped Gauss–Newton but without the need to solve linear systems (Kachhadiya, 26 Nov 2025, Kachhadiya, 13 May 2026). Proven results show the difference from true Gauss–Newton is bounded in terms of the JCP-controlled composition error and local conditioning (Kachhadiya, 13 May 2026).

4. Empirical Results and Benchmarks

Extensive benchmarking on synthetic and PDE-constrained inverse problems demonstrates the efficacy of Deceptron+D-IPG:

• On Heat-1D recovery: median 3 D-IPG iterations vs. 49 for gradient descent (x-GD), and 3 for Gauss–Newton, but D-IPG inference is 10–26× faster per tolerance (Kachhadiya, 26 Nov 2025).
• On Damped Oscillator: D-IPG achieves 2–3× lower iterations than x-GD and competes in cost with Gauss–Newton.
• On a suite of 2D/3D PDEs (Heat, Darcy, Advection-diffusion, Allen–Cahn, Navier–Stokes): D-IPG (+JCP) yields mean success rates of 94.8% versus 17.3% (GN) and 65.5% (LM), and achieves up to 77× faster inference cost (Kachhadiya, 13 May 2026).
• Ablation studies show omitting JCP sharply degrades reliability (e.g., Allen–Cahn-2D SR drops from 100% to 16.25%). JCP consistently correlates with improved reproducibility and convergence.

Empirical diagnostics include the RJCP measure, which tracks the inverse-consistency defect along solver trajectories and provides a reliability indicator (Kachhadiya, 13 May 2026).

5. Theoretical Analysis and Guarantees

The Deceptron framework provides first-order equivalence between D-IPG and (damped) Gauss–Newton steps under exact local pseudoinverse geometry. Explicit theorem statements prove:

• The local D-IPG update approximates the Gauss–Newton step up to a term proportional to the composition error $y^\star$2.
• The deviation from Gauss–Newton is further controlled by the inverse surrogate’s conditioning and the magnitude of the residual (Kachhadiya, 13 May 2026).
• Informally, this reproduces (near-) second-order directionality but amortized through a learned reverse map, eliminating the need for matrix decompositions or linear solves.

6. Architectural and Implementation Aspects

• MLP and residual CNN architectures for $y^\star$3 and $y^\star$4 are employed, with architectural symmetry and shallow depth.
• All Jacobians and their vector products (required for JCP/RJCP) are computed using automatic differentiation (e.g., JVP/VJP).
• In 2D spatial tasks, DeceptronNet v0 unrolls a learned corrector using a U-Net backbone; this achieves RMSE 0.0640 with just 6 steps (versus 69 for LM and 80 for GD) on 2D deblurring (Kachhadiya, 26 Nov 2025).
• The module is lightweight, easily integrating as a plugin for arbitrary differentiable forward surrogates.

7. Significance, Limitations, and Future Directions

The Deceptron module enables substantial acceleration of inverse problem solvers, frequently matching or surpassing second-order methods in iteration count, but at far lower computation cost. It also provides runtime interpretable diagnostics (RJCP) and is robust across a wide set of PDE-based benchmarks. Limitations include its locality—performance depends on surrogate fidelity and local invertibility; global nonidentifiability and poor forward models remain challenging (Kachhadiya, 26 Nov 2025, Kachhadiya, 13 May 2026).

Proposed extensions include:

• Multi-scale DeceptronNet variants for spatially-structured problems
• Integration with PINNs for physics-constrained inverse acceleration
• Adaptive JCP control using runtime diagnostics
• Application to high-dimensional or real-world systems identification

The Deceptron represents a milestone in learned inversion for physics-based machine learning, systematically combining geometric consistency with practical speed and generality (Kachhadiya, 26 Nov 2025, Kachhadiya, 13 May 2026).