Partial differential equations (PDEs) are the backbone of many models in geosciences and engineering—they describe how things like heat, fluid, or pollutants change over space and time. Recently, deep learning methods like Neural Operators have shown promise in solving these equations quickly. However, these methods fall short when it comes to understanding how sensitive the system is to changes in input conditions (a key need in science and engineering), adapting to new scenarios, or solving inverse problems (where we work backward from observations to causes). In this study, we introduce a new approach called Sensitivity-Constrained Neural Operators (SC-NO) that improves on these weaknesses. Our method teaches the model not just to get the right answer, but to also respond to input changes in the right way. It works even when the number of input variables is very large—over ten thousand—and it handles complex cases with irregular shapes and turbulent behavior. Once trained, SC-NO can predict outcomes quickly, even in new situations it hasn’t seen before, and its predictions stay accurate much longer than previous models. These improvements come with only a modest increase in training time. We demonstrate the method’s versatility across many types of equations relevant to environmental science.