The Rise of Differentiable Testing

Author: Denis Avetisyan

A new framework, DEFT, uses gradient-based optimization to dramatically improve the detection of hard-to-find faults in integrated circuits.

The DEFT framework establishes a differentiable path for reasoning about physical systems, enabling gradient-based optimization of control policies directly within the physics engine by representing continuous dynamics as <span class="katex-eq" data-katex-display="false"> \dot{x} = f(x, u) </span>, where <span class="katex-eq" data-katex-display="false"> x </span> denotes the system state and <span class="katex-eq" data-katex-display="false"> u </span> represents the control input, thus bridging the gap between learned control and provable system stability. — The DEFT framework establishes a differentiable path for reasoning about physical systems, enabling gradient-based optimization of control policies directly within the physics engine by representing continuous dynamics as $\dot{x} = f(x, u)$ , where $x$ denotes the system state and $u$ represents the control input, thus bridging the gap between learned control and provable system stability.

DEFT leverages a differentiable ATPG approach and a custom CUDA kernel to achieve significant gains in fault coverage and scalability compared to traditional test pattern generation methods.

As integrated circuit complexity escalates, so too does the challenge of generating effective test patterns, particularly for hard-to-detect faults. This paper introduces DEFT (Differentiable Automatic Test Pattern Generation), a novel approach that reformulates automatic test pattern generation as a continuous optimization problem solvable via gradient descent. By aligning continuous objectives with discrete fault detection semantics and leveraging a custom CUDA kernel, DEFT achieves significant improvements in fault coverage and scalability. Could this differentiable framework represent a paradigm shift in how we approach test generation for increasingly complex digital systems?

The Escalating Challenge of Fault Detection

As integrated circuit designs escalate in complexity, traditional Automatic Test Pattern Generation (ATPG) methods are increasingly challenged to thoroughly verify their functionality. This struggle isn’t simply due to the sheer number of components, but critically, the emergence of hard-to-detect (HTD) faults. These faults, often masked by redundant logic or subtle timing interactions, evade standard testing procedures designed for more obvious defects. Consequently, achieving high fault coverage – a key metric of testing effectiveness – becomes significantly more difficult, demanding substantial computational resources and potentially leaving subtle, reliability-threatening flaws undetected during manufacturing. The limitations of conventional ATPG in addressing HTD faults represent a growing concern as designs shrink and become more intricate, necessitating advanced techniques to ensure product quality and dependability.

Automatic Test Pattern Generation (ATPG) relies on comprehensive fault coverage to ensure the reliability of integrated circuits; however, an increasing number of hard-to-detect (HTD) faults present a substantial obstacle to achieving this goal. Fault coverage, expressed as the percentage of potential defects a test can identify, directly correlates with product quality; a higher percentage indicates a more robust and dependable design. HTD faults, by their nature, evade standard testing procedures due to factors like redundancy, masking effects from other faults, and limited controllability or observability within the circuit. Consequently, even sophisticated ATPG algorithms struggle to generate tests that reliably expose these defects, potentially leaving subtle, yet critical, flaws undetected and impacting long-term device performance. Addressing HTD faults, therefore, is paramount in modern semiconductor testing, requiring innovative techniques to augment traditional ATPG methods and bolster confidence in product integrity.

Despite the widespread use of commercial Automatic Test Pattern Generation (ATPG) tools in semiconductor testing, complex hard-to-detect (HTD) faults often present significant challenges. These tools, while generally effective at identifying common defects, struggle with faults masked by design intricacies or redundancy, requiring substantially more computational resources and test patterns to achieve acceptable fault coverage. This limitation directly translates into increased testing costs, as longer test times and greater pattern storage are needed. More critically, incomplete detection of HTD faults can lead to unreliable devices reaching the market, potentially causing field failures and impacting product longevity, thereby raising significant quality and safety concerns for manufacturers and consumers alike.

DEFT demonstrates comparable fault detection rates to a commercial ATPG tool, as shown by the similar number of detected faults on the y-axis.

Introducing DEFT: A Differentiable Approach to ATPG

DEFT, or Differentiable Automatic Test Pattern Generation, introduces a new approach to ATPG by integrating principles from differentiable programming. Traditional ATPG methods operate on discrete values, limiting the applicability of gradient-based optimization algorithms. DEFT overcomes this limitation by formulating the test pattern generation process within a differentiable framework. This allows for the use of techniques like backpropagation to efficiently search for test patterns that detect faults in integrated circuits. The core innovation lies in representing the normally discrete ATPG variables-input stimuli and fault models-in a continuous space, enabling gradient descent and related optimization methods to be applied directly to the test generation problem.

Traditional Automatic Test Pattern Generation (ATPG) methods operate on discrete logic values (0 or 1), precluding the application of gradient-based optimization algorithms. DEFT overcomes this limitation by reformulating ATPG as a continuous optimization problem; this is achieved by representing test vectors as continuous values rather than binary assignments. This continuous representation allows for the calculation of gradients, enabling the use of techniques like stochastic gradient descent and backpropagation to efficiently search for test patterns that maximize fault coverage. Consequently, DEFT can leverage the well-established infrastructure and advancements in differentiable programming to address ATPG challenges previously intractable with discrete approaches.

Continuous relaxation is a core component of the DEFT framework, addressing the inherent limitations of discrete ATPG methods. Traditionally, ATPG involves searching a discrete solution space of boolean input vectors; DEFT instead transforms these binary variables into continuous values. This is achieved by representing logic signals as real numbers between 0 and 1, and logic gates are modeled with differentiable functions approximating their boolean behavior. This allows the ATPG problem, previously defined by discrete constraints, to be expressed as a continuous optimization problem with a smooth, differentiable objective function. Consequently, gradient-based optimization algorithms, such as stochastic gradient descent, can be applied to efficiently search for test patterns that maximize the probability of fault detection.

DEFT's reparameterization technique outperforms naive continuous relaxation by enabling efficient gradient estimation for discrete actions. — DEFT’s reparameterization technique outperforms naive continuous relaxation by enabling efficient gradient estimation for discrete actions.

Key Technical Innovations within DEFT

DEFT utilizes Gumbel-Softmax reparameterization to enable gradient-based optimization through discrete sampling. This technique allows for differentiable approximation of categorical variables, facilitating backpropagation where direct differentiation is not possible. Further enhancing optimization efficiency, DEFT incorporates a custom CUDA kernel designed to accelerate value propagation during the forward and backward passes. This kernel is specifically optimized for the computational demands of the DEFT algorithm, resulting in significant performance gains compared to standard implementations.

A custom CUDA kernel was implemented to accelerate value propagation within the DEFT framework. Benchmarking indicates this kernel achieves a speedup ranging from 4.6x to 26x when compared to an equivalent implementation utilizing PyTorch and the Deep Graph Library (DGL). This performance gain is attributable to the kernel’s optimized memory access patterns and parallel processing capabilities on GPU hardware, significantly reducing computational time for large-scale graph processing tasks.

Gradient Normalization is implemented within DEFT to address instability during training, specifically mitigating the issue of exploding gradients. This technique rescales gradients based on their L2 norm, preventing excessively large gradient values from disrupting the optimization process. By clipping or normalizing these gradients, the system ensures that weight updates remain within a manageable range, thereby stabilizing learning and promoting convergence, particularly in deep networks where gradients can compound across multiple layers. The process involves calculating the norm of the gradient vector for each parameter and then dividing the gradient by this norm, effectively constraining its magnitude.

DEFT efficiently propagates continuous values using a custom CUDA kernel for accelerated computation.

Demonstrating DEFT’s Impact on Industrial Benchmarks

Evaluations of DEFT on industry-standard benchmarks, specifically Non-Cacheable Units (NCU) and MAC cells, reveal substantial gains in fault coverage when contrasted with conventional automated test pattern generation (ATPG) techniques. This improved performance isn’t merely incremental; DEFT consistently identifies a greater proportion of potential defects, indicating a more robust and reliable testing process. The system’s efficacy is demonstrated by its ability to detect elusive, hard-to-detect faults that often escape traditional methods, ultimately leading to higher quality and more dependable integrated circuits.

Evaluations reveal that DEFT substantially enhances the detection of particularly elusive, hard-to-detect (HTD) faults within integrated circuits. Compared to a leading commercial Automatic Test Pattern Generation (ATPG) tool, DEFT achieves an average improvement ranging from 21.1% to 48.9% in identifying these critical defects. This represents a significant advancement in test efficiency, as HTD faults often indicate underlying manufacturing or design weaknesses that could lead to field failures. The enhanced detection rate suggests DEFT’s algorithms are more effective at stimulating and observing the subtle electrical characteristics indicative of these complex faults, leading to more robust and reliable chip designs.

Evaluations on Non-Cacheable Unit (NCU) benchmarks reveal a significant efficiency advantage for DEFT in automated testing pattern generation. The system achieved complete fault coverage utilizing 116 test patterns, a notable reduction of 96 patterns compared to the leading commercial ATPG tool currently in use. This represents a substantial 17.8% decrease in the number of patterns required to ensure thorough testing, translating directly into reduced test times and lower testing costs for industrial applications. The ability to achieve full coverage with fewer patterns underscores DEFT’s optimized approach to fault detection and highlights its potential for streamlining manufacturing processes and improving product reliability.

Evaluations using Multiplier-Accumulator (MAC) cells reveal a substantial enhancement in fault detection capabilities with DEFT. The tool successfully identified over 800 faults within these complex circuits, a figure representing a significant 40% increase compared to the leading commercial Automated Test Pattern Generation (ATPG) tool. This improved detection rate suggests a heightened ability to uncover subtle defects that might otherwise escape traditional testing methods, potentially leading to more reliable hardware designs and a reduction in field failures for systems heavily reliant on MAC cell operations – such as those found in digital signal processing and machine learning applications.

Beyond enhanced fault detection, DEFT distinguishes itself through remarkable efficiency in test pattern generation. The methodology achieves a substantial 30.3% reduction in bit-ratio – a measure of the total number of bits required to represent the test patterns – while simultaneously improving fault coverage. This reduction translates directly to decreased testing time and lower storage requirements for test patterns, offering significant cost savings in manufacturing environments. The ability to accomplish more comprehensive testing with fewer patterns represents a critical advancement, particularly for complex integrated circuits where test data volume is a growing concern and efficient testing is paramount to maintaining production throughput.

DEFT identifies a stuck-at-1 fault at net n1 by visualizing gradient flow <span class="katex-eq" data-katex-display="false">→</span> (green) and highlighting the copied fanout cone introduced by the fault (red). — DEFT identifies a stuck-at-1 fault at net n1 by visualizing gradient flow $\to$ (green) and highlighting the copied fanout cone introduced by the fault (red).

Future Directions and the Promise of Differentiable ATPG

Future investigations are poised to integrate Differentiable Exact Testability (DEFT) with the adaptive learning capabilities of reinforcement learning. This synergistic approach aims to transcend the limitations of current search algorithms by enabling DEFT to learn optimal strategies for navigating the complex design space of Automatic Test Pattern Generation (ATPG). By framing the test generation process as a reinforcement learning problem, the system can iteratively refine its decision-making, prioritizing promising search paths and efficiently identifying critical faults. Such integration promises not only to accelerate test generation but also to enhance the quality of generated test patterns, leading to more robust and reliable chip designs, even in the face of increasingly intricate fault models and design complexities.

The true potential of Differentiable Fault Test (DEFT) hinges on its scalability to address the intricate challenges presented by contemporary integrated circuit designs and increasingly sophisticated fault models. Future investigations must move beyond simplified scenarios to encompass designs with billions of transistors and fault models that accurately reflect real-world failure mechanisms, including bridging faults, delay defects, and open transitions. Successfully adapting DEFT to these complex landscapes will necessitate innovations in memory management, parallelization strategies, and gradient optimization techniques, ultimately determining its viability as a cornerstone of next-generation automatic test pattern generation (ATPG) systems and its ability to meet the demands of ever-shrinking nanometer technologies.

The advent of Differentiable Fault Test (DEFT) signals a paradigm shift in Automated Test Pattern Generation (ATPG), moving beyond traditional, discrete search algorithms. By framing fault diagnosis as an optimization problem and leveraging the principles of differentiable programming, DEFT enables the use of gradient-based methods to efficiently navigate the complex design space of modern integrated circuits. This approach not only accelerates the test generation process but also offers the potential to address the escalating challenges posed by increasingly intricate chip architectures and sophisticated fault models. Consequently, a new generation of ATPG tools, built upon this foundation, promises to deliver significantly improved test coverage, reduced test times, and enhanced diagnostic capabilities, ultimately contributing to more reliable and cost-effective chip manufacturing.

DEFT draws an analogy between neural network training and differentiable automatic test pattern generation (ATPG), where green boxes indicate trainable components analogous to network weights and blue boxes represent fixed components similar to the testbench in ATPG.

The pursuit of robust fault detection, as exemplified by DEFT, aligns with a mathematician’s demand for demonstrable truth. G. H. Hardy once stated, “The essence of mathematics is its economy.” This sentiment resonates deeply within the framework presented; DEFT’s differentiable approach, unlike traditional ATPG methods, doesn’t rely on exhaustive search but rather on a continuous relaxation and gradient-based optimization. This offers an inherently more economical path to achieving high fault coverage, especially for hard-to-detect (HTD) faults. The custom CUDA kernel further optimizes this process, demonstrating that elegant solutions often stem from mathematical principles applied to practical engineering challenges.

What Lies Ahead?

The presented work, while demonstrating a quantifiable improvement in automated test pattern generation, merely shifts the locus of difficulty. The core challenge isn’t simply finding a test; it’s establishing a provable correspondence between the test and the elimination of a fault. DEFT, by relying on gradient-based optimization within a continuous relaxation, sidesteps the rigorous demand for deterministic truth. While the empirical results are compelling, the question persists: how much of the observed improvement is genuine fault coverage, and how much is a consequence of the optimization converging on a locally minimal, yet functionally insufficient, solution?

Future efforts must address this ambiguity. A critical extension would involve incorporating formal verification techniques to validate the generated test patterns, offering a guarantee of fault detection beyond statistical confidence. Furthermore, the reliance on a custom CUDA kernel, while currently providing a performance advantage, introduces a constraint on portability. A more elegant solution would be to express the underlying computations in a manner amenable to automatic differentiation by established frameworks, sacrificing some immediate speed for broader applicability and maintainability.

Ultimately, the pursuit of fault detection is a search for mathematical certainty in a domain often characterized by approximation. The true measure of progress will not be simply achieving higher fault coverage numbers, but rather establishing a framework where the correctness of a test can be demonstrably proven, not merely empirically observed.

Original article: https://arxiv.org/pdf/2512.23746.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/