[NB] add function and algorithm differentiation (#8644)

* [NB] add function and algorithm differentiation

 - algorithms and statements are differentiated properly
 - functions and function bodys are differentiated
   - recursion is handled
   - multiple occurences are handled
   - ToDo: strip interface variables such that only continuous are differentiated

* [NB] improve function differentiation

  1. if the function is builtin -> use hardcoded logic
  2. if the function is not builtin -> check if there is a 'fitting' derivative defined.
    - 'fitting' means that all the zeroDerivative annotations have to hold
  2.1 fitting function found -> use it
  2.2 fitting function not found -> differentiate the body of the function
  ToDo: respect the 'order' of the derivative when differentiating!

OMCompiler/Compiler/NBackEnd/Classes/NBVariable.mo

@@ -305,11 +305,11 @@ public
     output Boolean b;
   algorithm
     b := match Pointer.access(var)
-      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.DISCRETE_STATE())) then false;
-      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.DISCRETE())) then false;
-      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.PREVIOUS())) then false;
-      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.PARAMETER())) then false;
-      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.CONSTANT())) then false;
+      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.DISCRETE_STATE()))  then false;
+      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.DISCRETE()))        then false;
+      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.PREVIOUS()))        then false;
+      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.PARAMETER()))       then false;
+      case Variable.VARIABLE(backendinfo = BackendExtension.BACKEND_INFO(varKind = BackendExtension.CONSTANT()))        then false;
       else true;
     end match;
   end isContinuous;
@@ -818,21 +818,19 @@ public
     _ := match ComponentRef.node(cref)
       local
         InstNode qual;
-        Pointer<Variable> old_var_ptr;
         Variable var;
-      case qual as InstNode.VAR_NODE()
-        algorithm
-          // get the variable pointer from the old cref to later on link back to it
-          old_var_ptr := BVariable.getVarPointer(cref);
-          // prepend the seed str and the matrix name and create the new cref_DIFF_DIFF
-          qual.name := PARTIAL_DERIVATIVE_STR + "_" + name;
-          cref := ComponentRef.append(cref, ComponentRef.fromNode(qual, ComponentRef.scalarType(cref)));
-          var := fromCref(cref);
-          // update the variable to be a jac var and pass the pointer to the original variable
-          // ToDo: tmps will get JAC_DIFF_VAR !
-          var.backendinfo := BackendExtension.BackendInfo.setVarKind(var.backendinfo, BackendExtension.JAC_VAR());
-          // create the new variable pointer and safe it to the component reference
-          (var_ptr, cref) := makeVarPtrCyclic(var, cref);
+
+      // regular case for jacobians
+      case qual as InstNode.VAR_NODE() algorithm
+        // prepend the seed str and the matrix name and create the new cref_DIFF_DIFF
+        qual.name := PARTIAL_DERIVATIVE_STR + "_" + name;
+        cref := ComponentRef.append(cref, ComponentRef.fromNode(qual, ComponentRef.scalarType(cref)));
+        var := fromCref(cref);
+        // update the variable to be a jac var and pass the pointer to the original variable
+        // ToDo: tmps will get JAC_DIFF_VAR !
+        var.backendinfo := BackendExtension.BackendInfo.setVarKind(var.backendinfo, BackendExtension.JAC_VAR());
+        // create the new variable pointer and safe it to the component reference
+        (var_ptr, cref) := makeVarPtrCyclic(var, cref);
       then ();
 
       else algorithm
@@ -841,6 +839,28 @@ public
     end match;
   end makePDerVar;
 
+  function makeFDerVar
+      "Creates a function derivative cref. Used in NBDifferentiation
+    for differentiating body vars of a function."
+    input output ComponentRef cref    "old component reference to new component reference";
+  algorithm
+    cref := match ComponentRef.node(cref)
+      local
+        InstNode qual;
+
+      // for function differentiation (crefs are not lowered and only known locally)
+      case qual as InstNode.COMPONENT_NODE() algorithm
+        // prepend the seed str, matrix name locally not needed
+        qual.name := FUNCTION_DERIVATIVE_STR + "_" + qual.name;
+        cref := ComponentRef.fromNode(qual, ComponentRef.scalarType(cref));
+     then cref;
+
+      else algorithm
+        Error.addMessage(Error.INTERNAL_ERROR,{getInstanceName() + " failed for " + ComponentRef.toString(cref)});
+      then fail();
+    end match;
+  end makeFDerVar;
+
   function makeStartVar
     "Creates a start variable pointer from a cref. Used in NBInitialization.
     e.g: angle -> $START.angle"

OMCompiler/Compiler/NBackEnd/Modules/3_Post/NBJacobian.mo

@@ -537,7 +537,6 @@ protected
       jacobianHT      = optHT, // seed and temporary cref hashtable
       diffType        = NBDifferentiate.DifferentiationType.JACOBIAN,
       funcTree        = funcTree,
-      diffedFunctions = AvlSetPath.new(),
       scalarized      = seedCandidates.scalarized
     );
 
@@ -546,6 +545,7 @@ protected
 
     (diffed_eqn_lst, diffArguments) := Differentiate.differentiateEquationPointerList(listReverse(eqn_lst), diffArguments, idx, name, getInstanceName());
     diffedEquations := EquationPointers.fromList(diffed_eqn_lst);
+    funcTree := diffArguments.funcTree;
 
     // create equation data for jacobian
     // ToDo: split temporary and auxiliares once tearing is applied

OMCompiler/Compiler/NBackEnd/Modules/3_Post/NBSolve.mo

@@ -385,7 +385,6 @@ public
         jacobianHT      = NONE(),
         diffType        = NBDifferentiate.DifferentiationType.SIMPLE,
         funcTree        = funcTree,
-        diffedFunctions = AvlSetPath.new(),
         scalarized      = false
       );
       (derivative, diffArgs) := Differentiate.differentiateExpressionDump(residual, diffArgs, getInstanceName());

OMCompiler/Compiler/NBackEnd/Util/NBBackendUtil.mo

@@ -638,5 +638,24 @@ public
     end if;
   end isOnlyTimeDependentFold;
 
+  function isContinuous
+    input Expression exp;
+    output Boolean b;
+  algorithm
+    b := Expression.fold(exp, isContinuousFold, true);
+  end isContinuous;
+
+  function isContinuousFold
+    input Expression exp;
+    input output Boolean b;
+  algorithm
+    if b then
+      b := match exp
+        case Expression.CREF() then BVariable.checkCref(exp.cref, BVariable.isContinuous);
+        else true;
+      end match;
+    end if;
+  end isContinuousFold;
+
   annotation(__OpenModelica_Interface="backend");
 end NBBackendUtil;