Train Once for Budget-Adaptive Deployment: Learning Nested Sensor Layouts for Physical Field Reconstruction

Accurately reconstructing physical fields from sparse observations depends on model capability and sensor placement. In practice, sensor placement must also accommodate budgets that may change with resource availability or sensing requirements. However, existing placement methods typically optimize layouts for a single sensor budget, so budget changes require reoptimizing all sensor locations. We therefore propose Budget-Adaptive Sensor Placement (BASP), a differentiable framework that learns nested layouts across budgets in a single training run. BASP first encodes the physical field prior into spatial features, supplementing reconstruction feedback with explicit structural guidance. Then, ordered sensor queries sequentially generate continuous coordinates by combining cross attention to these features with spatial suppression, forming nested sensor layouts. During training, a pretrained, frozen reconstructor jointly supervises reconstruction at a randomly sampled budget and the maximum budget, encouraging early sensors to remain informative across layout sizes. Experiments on four datasets with four reconstructors show BASP achieves lower reconstruction errors than independent coordinate optimization, random and uniform sampling, D-, A-, and E-optimal placement in most settings. Beyond accuracy gains, BASP supports multiple budgets in one training run and enables rapid budget adjustment without relocating retained sensors.