PhysEDA: Physics-Aware EDA Framework

• The paper introduces PhysEDA, a physics-aware framework that integrates a Manhattan-distance decay bias via PSLA and PBRS to overcome quadratic complexity and overfitting.
• It leverages a shared physical prior to infuse spatial inductive bias into both model architecture and training, thereby enhancing scalability and cross-scale generalization in EDA tasks.
• Empirical results demonstrate significant performance improvements in sparse-reward reinforcement learning, decoupling-capacitor and macro placement, and IR-drop prediction under data-scarce conditions.

Searching arXiv for the named paper and closely related EDA learning frameworks. PhysEDA is a physics-aware learning framework for electronic design automation that incorporates a shared physical prior into both model architecture and training. It is introduced in “PhysEDA: Physics-Aware Learning Framework for Efficient EDA With Manhattan Distance Decay” (Yang, 11 May 2026) to address two bottlenecks in learning-based EDA: the quadratic scaling of vanilla attention and the tendency of data-scarce models to overfit statistical noise while amplifying weak long-range correlations against the underlying physics. The framework is evaluated on decoupling-capacitor placement, macro placement, and IR-drop prediction, and its central premise is that many electrical and routing interactions decay approximately exponentially with Manhattan distance. PhysEDA operationalizes that prior through Physics-Structured Linear Attention (PSLA) and Potential-Based Reward Shaping (PBRS), using the same kernel both as an architectural inductive bias and as a shaping potential for sparse-reward reinforcement learning (Yang, 11 May 2026).

1. Definition and scope

PhysEDA is formulated for EDA tasks that are spatial, grid-structured, and physically constrained. The paper studies three scenarios: decoupling-capacitor placement, macro placement, and IR-drop prediction (Yang, 11 May 2026). In each case, the method assumes that pairwise interactions are not arbitrary, but follow locality patterns aligned with chip physics or routing structure.

The framework comprises two components. The first is Physics-Structured Linear Attention (PSLA), which embeds a Manhattan-distance decay bias into linear attention while preserving linear asymptotic complexity. The second is Potential-Based Reward Shaping (PBRS), which uses the same decay kernel to provide dense reward signals in sparse-reward reinforcement learning while preserving the optimal policy through the policy-invariance theorem (Yang, 11 May 2026).

The unifying idea is that one physical prior can simultaneously improve representational bias, speed, memory use, and RL exploration. This suggests that PhysEDA is not organized as a task-specific heuristic, but as a shared inductive-bias layer spanning architecture and training.

2. Physical prior: Manhattan-distance decay

The paper derives the relevant prior from power-delivery-network physics. For power-delivery and IR-drop settings, the transfer impedance from node $j$ to node $i$ decays approximately exponentially with Manhattan distance: $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$ with $\alpha_x,\alpha_y>0$ (Yang, 11 May 2026).

A key structural feature is separability along horizontal and vertical coordinates. That separability is what makes the prior usable inside linear attention without constructing a full pairwise distance matrix. The same functional form is also justified for macro placement by routing locality: closer macros are more likely to share nets (Yang, 11 May 2026). Although the physical interpretation differs across tasks, the paper treats the exponential Manhattan-decay pattern as a common inductive bias across all three settings.

This prior is presented as an alternative to learning spatial structure entirely from data. The paper argues that purely data-driven models may require many examples to rediscover the same locality pattern and may overfit weak long-range correlations when training data are limited (Yang, 11 May 2026). A plausible implication is that PhysEDA is most useful when the target task has genuine spatial physics but insufficient data to infer it robustly.

3. Physics-Structured Linear Attention

PhysEDA starts from standard linear attention, which uses a positive feature map $\phi$ and avoids explicit formation of the $L\times L$ attention matrix: $\mathrm{LinAttn}(Q,K,V)=\frac{\phi(Q)[\phi(K)^\top V]}{\phi(Q)\phi(K)^\top\mathbf{1}.$ This reduces the cost of softmax attention from $\mathcal{O}(L^2 d)$ to $\mathcal{O}(Ld^2)$, with memory reduced from $\mathcal{O}(L^2)$ attention storage to $i$0 (Yang, 11 May 2026).

PSLA injects the Manhattan prior as a multiplicative bias on query and key features: $i$1 with

$i$2

The effective pairwise weight is therefore

$i$3

This construction preserves content-based similarity while multiplying it by a physics-inspired positional decay. Because $i$4 and $i$5 are separable, PSLA does not need to construct the full pairwise Manhattan-distance matrix, which is the mechanism by which linear complexity is preserved (Yang, 11 May 2026).

The decay rates $i$6 are learned per head and per layer, initialized around $i$7 and constrained to $i$8 (Yang, 11 May 2026). The paper states that this allows different heads to specialize to different spatial scales. It also reports that learned $i$9 values become anisotropic when the task has directional asymmetry, which is consistent with the use of separate horizontal and vertical decay parameters.

A limitation is explicitly noted. The main PSLA implementation uses a rank-1 factorization that realizes a directional or signed exponential rather than the fully symmetric $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$0. The appendix gives a bidirectional prefix-sum construction for exact symmetric reconstruction, but the main experiments use the simpler rank-1 form because it is much faster and works well in practice (Yang, 11 May 2026).

4. Potential-Based Reward Shaping

PBRS addresses sparse-reward reinforcement learning, especially in decoupling-capacitor placement. In that setting, the simulator returns a meaningful reward only after all capacitors are placed, typically via a Kron-reduction-based terminal cost, so intermediate actions are effectively unlabeled (Yang, 11 May 2026).

The reward is modified using a potential function $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$1: $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$2 The paper invokes the theorem from Ng et al., according to which the shaped return differs from the original return only by an endpoint term,

$|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$3

so the optimal policy is unchanged (Yang, 11 May 2026). PBRS is therefore presented as a mechanism for accelerating learning without altering the optimum.

For decoupling-capacitor placement, the potential is built directly from the same exponential kernel: $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$4 Here $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$5 is the set of already placed capacitors, $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$6 is the probe port, the first term encourages capacitors near the probe, and the second discourages clustering and diminishing returns (Yang, 11 May 2026).

For macro placement, the shaping potential is connectivity-based: $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$7 The paper notes that this is physically aligned with a negative-HPWL objective, though expressed in a connectivity-aware form (Yang, 11 May 2026).

The shaping term is annealed during training by a cosine-decayed schedule: $|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$8 This makes the prior strong early in training and weaker later. The stated purpose is to guide exploration initially while allowing later optimization to be driven more by the simulator reward (Yang, 11 May 2026).

5. Tasks and reported performance

PhysEDA is evaluated on three EDA problems: decoupling-capacitor placement, macro placement, and IR-drop prediction (Yang, 11 May 2026). The tasks differ in objective and model context, but all are used to test the same Manhattan-decay prior.

Task	Setting	Reported outcome
Decoupling-capacitor placement	$|Z_{\mathrm{tr}(i,j,f)| \approx Z_0(f)\exp\!\bigl(-\alpha_x|x_i-x_j|-\alpha_y|y_i-y_j|\bigr),$9 and $\alpha_x,\alpha_y>0$0 grids	PSLA and PBRS improve supervised, RL, and zero-shot cross-scale performance
Macro placement	ISPD 2005 adaptec1 with 452 macros	PSLA improves Decision-Transformer pretraining; PBRS improves RL fine-tuning
IR-drop prediction	CircuitNet, UNet bottleneck replaced by PSLA	Slightly worse in-distribution, better under cross-design and cross-architecture shift

For supervised DPP on $\alpha_x,\alpha_y>0$1, PSLA attains $\alpha_x,\alpha_y>0$2 versus $\alpha_x,\alpha_y>0$3 for DevFormer, a reported 5.5% improvement over the quadratic baseline (Yang, 11 May 2026). The paper emphasizes that non-physics linear-attention baselines underperform DevFormer, which is used to argue that linearization alone is insufficient.

For DPP reinforcement learning on $\alpha_x,\alpha_y>0$4 with REINFORCE, the reported results are $\alpha_x,\alpha_y>0$5 for vanilla DevFormer, $\alpha_x,\alpha_y>0$6 for PSLA, and $\alpha_x,\alpha_y>0$7 when PBRS is added (Yang, 11 May 2026). The paper summarizes this as a 49.1% gain from PSLA over DevFormer in the hard RL setting, with PBRS providing another 10.8% relative improvement over non-shaped PSLA. On the easier $\alpha_x,\alpha_y>0$8 task, the gain is only 1.5% without shaping, and PBRS largely removes the difference.

Zero-shot cross-scale transfer is a major result. Training on $\alpha_x,\alpha_y>0$9 and testing on $\phi$0 with no retraining, the paper reports 56.8% relative improvement for PSLA over DevFormer (Yang, 11 May 2026). The stated explanation is that Manhattan decay is scale-invariant in coordinate space, whereas learned relative position embeddings can fail when grid resolution changes.

For macro placement during Decision-Transformer pretraining on adaptec1, the reported HPWL values are 910,967 for the GPT DT baseline, 837,584 for PSLA DT, and 801,588 for Connectivity PSLA DT, corresponding to up to 12.0% improvement (Yang, 11 May 2026). During RL fine-tuning, GPT + PBRS logit bias reaches 693,999 HPWL compared with 733,794 for GPT REINFORCE, a 5.4% improvement.

For IR-drop prediction on CircuitNet, PSLA is reported to be slightly worse in-distribution on Vortex, about -1.9%, but better under distribution shift: +5.3% on cross-design and +5.4% on cross-architecture evaluation (Yang, 11 May 2026). The paper treats this as an important nuance rather than a universal gain.

6. Efficiency, ablations, and limitations

The efficiency results are tied directly to the linear-attention formulation. On $\phi$1 grids, PSLA yields about 14× inference speedup and 98.5% memory savings for the DPP encoder (Yang, 11 May 2026). At $\phi$2, softmax attention is reported to exceed GPU memory while PSLA still runs. These measurements are presented as practical consequences of replacing quadratic attention memory with the linear-attention factorization.

The ablation results support what the paper describes as a unified “data insufficiency” story (Yang, 11 May 2026). The main findings are that removing the decay hurts performance, Euclidean decay is worse than Manhattan decay, the narrow physically motivated $\phi$3 range matters more in DPP than in CircuitNet, and learned $\phi$4 values become anisotropic when the task exhibits directional asymmetry.

The paper also reports a boundary condition on when the prior helps. When the baseline already sees enough data, the prior can be redundant or slightly harmful; when data are scarce, shifted, or cross-scale, the prior helps substantially (Yang, 11 May 2026). This is most visible in sparse-reward RL, zero-shot cross-scale transfer, and shifted IR-drop evaluation. A plausible implication is that PhysEDA should be understood as a bias-variance tradeoff mechanism rather than as an unconditional improvement.

Two limitations are explicitly identified. First, the main PSLA formulation uses a directional rank-1 approximation rather than exact symmetric Manhattan decay. Second, PBRS applies only to online RL, not supervised learning (Yang, 11 May 2026). The IR-drop results add a third practical caveat: a physical prior can reduce flexibility when data are abundant enough for the baseline to learn the structure by itself.

7. Position within physics-aware machine learning for EDA

The central claim of PhysEDA is that EDA tasks are fundamentally spatial and constrained by real circuit physics, so relevant structure should be injected rather than rediscovered from scarce data (Yang, 11 May 2026). In this framework, the same Manhattan-distance prior is used twice: in attention, to improve scalability and spatial generalization, and in reward shaping, to improve sparse-reward exploration without changing the optimal policy.

This design links model efficiency and learning dynamics through a single inductive bias. PSLA addresses representational structure and computational scaling; PBRS addresses credit assignment and exploration. The empirical pattern reported across tasks is internally consistent: the less available data there are for inferring spatial structure, the more useful the prior becomes (Yang, 11 May 2026).

PhysEDA therefore occupies a specific position within learning-based EDA. It does not replace the underlying objectives of placement, routing, timing, or integrity analysis. Instead, it reparameterizes the learning problem so that architecture and training are aligned with an explicitly stated physical prior. Within the evidence reported in the paper, its strongest gains appear in sparse-reward reinforcement learning, cross-scale transfer, and distribution shift, while its weakest gains appear in settings where the baseline can already learn the relevant structure directly from abundant data (Yang, 11 May 2026).