Par­tial dif­fer­en­tial equa­tions can mod­el di­verse phys­ic­al phe­nom­ena in­clud­ing heat dif­fu­sion, in­com­press­ible flows, and elec­tro­stat­ic po­ten­tials. Giv­en a de­scrip­tion of an ob­ject’s bound­ary and in­teri­or, tra­di­tion­al meth­ods solve such PDEs by densely mesh­ing the in­teri­or and then solv­ing a large and sparse lin­ear sys­tem de­rived from this mesh. Re­cent grid-free solv­ers take an al­tern­at­ive ap­proach and avoid this com­plex­ity in ex­change for ran­dom­ness: they com­pute stochast­ic solu­tion es­tim­ates and gen­er­ally bear a strik­ing re­semb­lance to phys­ic­ally-based ren­der­ing al­gorithms.

In this art­icle, we de­vel­op al­gorithms tar­get­ing the in­verse form of this prob­lem: giv­en an already ex­ist­ing solu­tion of a PDE, we in­fer para­met­ers char­ac­ter­iz­ing the bound­ary and in­teri­or. In the grid-free set­ting, there are again sig­ni­fic­ant con­nec­tions to ren­der­ing, and we show how in­sights from both fields can be com­bined to com­pute un­biased de­riv­at­ive es­tim­ates that en­able gradi­ent-based op­tim­iz­a­tion. In this pro­cess, we en­counter new chal­lenges that must be ad­dressed to ob­tain prac­tic­al solu­tions. We in­tro­duce ac­cel­er­a­tion and vari­ance re­duc­tion strategies and show how to dif­fer­en­ti­ate branch­ing ran­dom walks in re­verse mode. We fi­nally demon­strate our ap­proach on both sim­u­lated data and a real-world elec­tric­al im­ped­ance tomo­graphy ex­per­i­ment, where we re­con­struct the po­s­i­tion of a con­duct­ing ob­ject from voltage meas­ure­ments taken in a sa­line-filled tank.