Inclusion Criteria:

          -  High FGFR mRNA expression levels (RNAscope score of ≥3; measurement is part of this
             protocol) in archival or fresh tumor biopsy specimen.

          -  Eastern Cooperative Oncology Group Performance Status (ECOG PS) of 0 or 1.

          -  At least 1 measurable lesion according to Response Evaluation Criteria in Solid Tumors
             (RECIST 1.1) in contrast enhanced (unless contraindicated) CT or MRI.

          -  Adequate bone marrow, liver and renal function.

          -  Glomerular filtration rate (GFR) ≥ 30 mL/min/1.73 m*2 according to the Modification of
             Diet in Renal Disease (MDRD) formula.

          -  Left ventricular ejection fraction (LVEF) equal to or above the lower limit of normal
             (LLN) at the institution.

          -  Life expectancy of at least 3 months.

          -  For the dose escalation part: Patients with histologically confirmed, locally advanced
             or metastatic solid tumors who are not candidates for or refuse standard therapy or
             whose disease progressed and for which standard anti-cancer treatment is no longer
             effective, excluding primary brain or spinal tumors. Patients who have been advised
             with all standard treatment options and still refuse them must be documented and can
             be allowed to enter the trial.

          -  For the dose expansion part: Patients with histologically confirmed, locally advanced
             or metastatic urothelial carcinoma (transitional cell carcinoma) including urinary
             bladder, renal pelvis, ureters, urethra who are not candidates for or refuse standard
             therapy or whose disease progressed and for which standard anticancer treatment is no
             longer effective. Patients who have been advised with all standard treatment options
             and still refuse them must be documented and can be allowed to enter the trial.

        Exclusion Criteria:

          -  Previous or concurrent cancer that is distinct from tumor for which the patient is
             enrolled in the study, except

               -  curatively treated cervical carcinoma in situ

               -  treated basal-cell carcinoma

               -  localized prostate cancer treated with curative intent and known absence of
                  prostate-specific antigen (PSA) relapse or incidental prostate cancer (T1/T2a,
                  Gleason score ≤ 6, and PSA ≤ 10 ng/mL undergoing active surveillance and
                  treatment-naïve)

               -  any cancer curatively treated > 3 years before planned start of study treatment.

          -  Ongoing or previous anti-cancer treatment within 4 weeks of study treatment start (or
             6 weeks for mitomycin C, nitrosoureas and monoclonal antibodies); with exceptions.

          -  Prior toxicity to anti-FGFR-directed or anti-PI3K-directed therapies leading to
             treatment discontinuation (previous exposure is allowed in other circumstances). If
             prior toxicity to anti-FGFR-directed or anti-PI3K-directed therapies leading to
             treatment discontinuation is different from the known safety profile of rogaratinib or
             copanlisib, enrollment is allowed.

          -  Symptomatic brain or meningeal metastatic tumors unless the patient is >6 months from
             definitive therapy, has no evidence of tumor growth on an imaging study and is
             clinically stable with respect to the tumor at the start of study treatment. Also the
             patient must not be undergoing acute steroid therapy or taper (chronic steroid therapy
             is acceptable provided that the dose is stable for one month prior to and following
             screening radiographic studies).

          -  History or current condition of an uncontrolled cardiovascular disease including
             congestive heart failure NYHA > Class 2, unstable angina (symptoms of angina at rest)
             or new-onset angina (within last 3 months) or myocardial infarction within past 6
             months and cardiac arrhythmias requiring anti-arrhythmic therapy (beta-blockers or
             digoxin are permitted).

          -  Active hepatitis B (HBV) or C (HCV) infection.

          -  Active clinically serious infections (≥ CTCAE v4.03 Grade 2).